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6482:{\displaystyle {\begin{aligned}\mathbf {U} &={\begin{bmatrix}\color {Green}0&\color {Blue}-1&\color {Cyan}0&\color {Emerald}0\\\color {Green}-1&\color {Blue}0&\color {Cyan}0&\color {Emerald}0\\\color {Green}0&\color {Blue}0&\color {Cyan}0&\color {Emerald}-1\\\color {Green}0&\color {Blue}0&\color {Cyan}-1&\color {Emerald}0\end{bmatrix}}\\\mathbf {\Sigma } &={\begin{bmatrix}3&0&0&0&\color {Gray}{\mathit {0}}\\0&{\sqrt {5}}&0&0&\color {Gray}{\mathit {0}}\\0&0&2&0&\color {Gray}{\mathit {0}}\\0&0&0&\color {Red}\mathbf {0} &\color {Gray}{\mathit {0}}\end{bmatrix}}\\\mathbf {V} ^{*}&={\begin{bmatrix}\color {Violet}0&\color {Violet}0&\color {Violet}-1&\color {Violet}0&\color {Violet}0\\\color {Plum}-{\sqrt {0.2}}&\color {Plum}0&\color {Plum}0&\color {Plum}0&\color {Plum}-{\sqrt {0.8}}\\\color {Magenta}0&\color {Magenta}-1&\color {Magenta}0&\color {Magenta}0&\color {Magenta}0\\\color {Orchid}0&\color {Orchid}0&\color {Orchid}0&\color {Orchid}1&\color {Orchid}0\\\color {Purple}-{\sqrt {0.8}}&\color {Purple}0&\color {Purple}0&\color {Purple}0&\color {Purple}{\sqrt {0.2}}\end{bmatrix}}\end{aligned}}}
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7293:{\displaystyle \mathbf {V} ^{*}={\begin{bmatrix}\color {Violet}0&\color {Violet}1&\color {Violet}0&\color {Violet}0&\color {Violet}0\\\color {Plum}0&\color {Plum}0&\color {Plum}1&\color {Plum}0&\color {Plum}0\\\color {Magenta}{\sqrt {0.2}}&\color {Magenta}0&\color {Magenta}0&\color {Magenta}0&\color {Magenta}{\sqrt {0.8}}\\\color {Orchid}{\sqrt {0.4}}&\color {Orchid}0&\color {Orchid}0&\color {Orchid}{\sqrt {0.5}}&\color {Orchid}-{\sqrt {0.1}}\\\color {Purple}-{\sqrt {0.4}}&\color {Purple}0&\color {Purple}0&\color {Purple}{\sqrt {0.5}}&\color {Purple}{\sqrt {0.1}}\end{bmatrix}}}
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23599:{\displaystyle {\begin{aligned}\sigma _{\pm }&={\sqrt {|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}\pm {\sqrt {{\bigl (}|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}{\bigr )}^{2}-|z_{0}^{2}-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}|^{2}}}}}\\&={\sqrt {|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}\pm 2{\sqrt {(\operatorname {Re} z_{0}z_{1}^{*})^{2}+(\operatorname {Re} z_{0}z_{2}^{*})^{2}+(\operatorname {Re} z_{0}z_{3}^{*})^{2}+(\operatorname {Im} z_{1}z_{2}^{*})^{2}+(\operatorname {Im} z_{2}z_{3}^{*})^{2}+(\operatorname {Im} z_{3}z_{1}^{*})^{2}}}}}\end{aligned}}}
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15865:{\displaystyle {\begin{bmatrix}\mathbf {V} _{1}^{*}\\\mathbf {V} _{2}^{*}\end{bmatrix}}\mathbf {M} ^{*}\mathbf {M} \,{\begin{bmatrix}\mathbf {V} _{1}&\!\!\mathbf {V} _{2}\end{bmatrix}}={\begin{bmatrix}\mathbf {V} _{1}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{1}&\mathbf {V} _{1}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{2}\\\mathbf {V} _{2}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{1}&\mathbf {V} _{2}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{2}\end{bmatrix}}={\begin{bmatrix}\mathbf {D} &0\\0&0\end{bmatrix}}.}
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18469:{\displaystyle {\begin{bmatrix}\mathbf {U} _{1}&\mathbf {U} _{2}\end{bmatrix}}{\begin{bmatrix}{\begin{bmatrix}\mathbf {} D^{\frac {1}{2}}&0\\0&0\end{bmatrix}}\\0\end{bmatrix}}{\begin{bmatrix}\mathbf {V} _{1}&\mathbf {V} _{2}\end{bmatrix}}^{*}={\begin{bmatrix}\mathbf {U} _{1}&\mathbf {U} _{2}\end{bmatrix}}{\begin{bmatrix}\mathbf {D} ^{\frac {1}{2}}\mathbf {V} _{1}^{*}\\0\end{bmatrix}}=\mathbf {U} _{1}\mathbf {D} ^{\frac {1}{2}}\mathbf {V} _{1}^{*}=\mathbf {M} ,}
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9286:{\displaystyle {\begin{aligned}\mathbf {M} ^{*}\mathbf {M} &=\mathbf {V} \mathbf {\Sigma } ^{*}\mathbf {U} ^{*}\,\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}=\mathbf {V} (\mathbf {\Sigma } ^{*}\mathbf {\Sigma } )\mathbf {V} ^{*},\\\mathbf {M} \mathbf {M} ^{*}&=\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}\,\mathbf {V} \mathbf {\Sigma } ^{*}\mathbf {U} ^{*}=\mathbf {U} (\mathbf {\Sigma } \mathbf {\Sigma } ^{*})\mathbf {U} ^{*}.\end{aligned}}}
6996:{\displaystyle {\begin{aligned}\mathbf {U} \mathbf {U} ^{*}&={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}=\mathbf {I} _{4}\\\mathbf {V} \mathbf {V} ^{*}&={\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{bmatrix}}=\mathbf {I} _{5}\end{aligned}}}
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16682:{\displaystyle \mathbf {U} _{1}\mathbf {D} ^{\frac {1}{2}}\mathbf {V} _{1}^{*}=\mathbf {M} \mathbf {V} _{1}\mathbf {D} ^{-{\frac {1}{2}}}\mathbf {D} ^{\frac {1}{2}}\mathbf {V} _{1}^{*}=\mathbf {M} (\mathbf {I} -\mathbf {V} _{2}\mathbf {V} _{2}^{*})=\mathbf {M} -(\mathbf {M} \mathbf {V} _{2})\mathbf {V} _{2}^{*}=\mathbf {M} ,}
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are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. The number of independent left and
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are used to estimate the most linearly quickly growing few perturbations to the central numerical weather prediction over a given initial forward time period; i.e., the singular vectors corresponding to the largest singular values of the linearized propagator for the global weather over that time
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produces left- and right- unitary singular matrices. This approach cannot readily be accelerated, as the QR algorithm can with spectral shifts or deflation. This is because the shift method is not easily defined without using similarity transformations. However, this iterative approach is very
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In applications it is quite unusual for the full SVD, including a full unitary decomposition of the null-space of the matrix, to be required. Instead, it is often sufficient (as well as faster, and more economical for storage) to compute a reduced version of the SVD. The following can be
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Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO. SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models.
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28547:{\displaystyle \mathbf {M} \psi =\mathbf {U} T_{f}\mathbf {V} ^{*}\psi =\sum _{i}\left\langle \mathbf {U} T_{f}\mathbf {V} ^{*}\psi ,\mathbf {U} e_{i}\right\rangle \mathbf {U} e_{i}=\sum _{i}\sigma _{i}\left\langle \psi ,\mathbf {V} e_{i}\right\rangle \mathbf {U} e_{i},}
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17735:{\displaystyle \mathbf {U} _{1}^{*}\mathbf {U} _{1}=\mathbf {D} ^{-{\frac {1}{2}}}\mathbf {V} _{1}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{1}\mathbf {D} ^{-{\frac {1}{2}}}=\mathbf {D} ^{-{\frac {1}{2}}}\mathbf {D} \mathbf {D} ^{-{\frac {1}{2}}}=\mathbf {I_{1}} ,}
16345:{\displaystyle {\begin{aligned}\mathbf {V} _{1}^{*}\mathbf {V} _{1}&=\mathbf {I} _{1},\\\mathbf {V} _{2}^{*}\mathbf {V} _{2}&=\mathbf {I} _{2},\\\mathbf {V} _{1}\mathbf {V} _{1}^{*}+\mathbf {V} _{2}\mathbf {V} _{2}^{*}&=\mathbf {I} _{12},\end{aligned}}}
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can be chosen to be both rotations with reflections, or both rotations without reflections. If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.
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in other fields) uses SVD to compute the optimal rotation (with respect to least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.
13178:{\displaystyle \mathbf {O} ={\underset {\Omega }{\operatorname {argmin} }}\|\mathbf {A} {\boldsymbol {\Omega }}-\mathbf {B} \|_{F}\quad {\text{subject to}}\quad {\boldsymbol {\Omega }}^{\operatorname {T} }{\boldsymbol {\Omega }}=\mathbf {I} ,}
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is exactly the rank of the matrix. Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described by a
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as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero
19971:{\displaystyle \nabla \sigma =\nabla \mathbf {u} ^{\operatorname {T} }\mathbf {M} \mathbf {v} -\lambda _{1}\cdot \nabla \mathbf {u} ^{\operatorname {T} }\mathbf {u} -\lambda _{2}\cdot \nabla \mathbf {v} ^{\operatorname {T} }\mathbf {v} }
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Euclidean inner product. For this reason, it is also called the operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator
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In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number"
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20127:{\displaystyle {\begin{aligned}\mathbf {M} \mathbf {v} _{1}&=2\lambda _{1}\mathbf {u} _{1}+0,\\\mathbf {M} ^{\operatorname {T} }\mathbf {u} _{1}&=0+2\lambda _{2}\mathbf {v} _{1}.\end{aligned}}}
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SVD has also been applied to reduced order modelling. The aim of reduced order modelling is to reduce the number of degrees of freedom in a complex system which is to be modeled. SVD was coupled with
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is the Jacobi transformation matrix that zeroes these off-diagonal elements. The iterations proceeds exactly as in the Jacobi eigenvalue algorithm: by cyclic sweeps over all off-diagonal elements.
14187:{\displaystyle \nabla \mathbf {x} ^{\operatorname {T} }\mathbf {M} \mathbf {x} -\lambda \cdot \nabla \mathbf {x} ^{\operatorname {T} }\mathbf {x} =2(\mathbf {M} -\lambda \mathbf {I} )\mathbf {x} .}
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20459:{\displaystyle {\begin{aligned}\mathbf {M} \mathbf {v} _{1}&=\sigma _{1}\mathbf {u} _{1},\\\mathbf {M} ^{\operatorname {T} }\mathbf {u} _{1}&=\sigma _{1}\mathbf {v} _{1}.\end{aligned}}}
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22035:). Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition.
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The Scale-Invariant SVD, or SI-SVD, is analogous to the conventional SVD except that its uniquely-determined singular values are invariant with respect to diagonal transformations of
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13782:{\displaystyle f:\left\{{\begin{aligned}\mathbb {R} ^{n}&\to \mathbb {R} \\\mathbf {x} &\mapsto \mathbf {x} ^{\operatorname {T} }\mathbf {M} \mathbf {x} \end{aligned}}\right.}
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28936:(which are compact, under some weak technical assumptions); it seems he was unaware of the parallel work on singular values of finite matrices. This theory was further developed by
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Depireux, D. A.; Simon, J. Z.; Klein, D. J.; Shamma, S. A. (March 2001). "Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex".
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3165:-dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction. See
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also arrived at the singular value decomposition for real square matrices in 1889, apparently independently of both
Beltrami and Jordan. Sylvester called the singular values the
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of the non-zero singular values is large making even the
Compact SVD impractical to compute. In such cases, the smallest singular values may need to be truncated to compute only
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16001:{\displaystyle \mathbf {V} _{1}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{1}=\mathbf {D} ,\quad \mathbf {V} _{2}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{2}=\mathbf {0} .}
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simple to implement, so is a good choice when speed does not matter. This method also provides insight into how purely orthogonal/unitary transformations can obtain the SVD.
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One-sided Jacobi algorithm is an iterative algorithm, where a matrix is iteratively transformed into a matrix with orthogonal columns. The elementary iteration is given as a
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Singular values are similar in that they can be described algebraically or from variational principles. Although, unlike the eigenvalue case, Hermiticity, or symmetry, of
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is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero
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may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.
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5836:{\displaystyle \mathbf {M} ={\begin{bmatrix}1&0&0&0&2\\0&0&3&0&0\\0&0&0&0&0\\0&2&0&0&0\end{bmatrix}}}
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interval. The output singular vectors in this case are entire weather systems. These perturbations are then run through the full nonlinear model to generate an
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26538:{\displaystyle \|\mathbf {M} \|={\sqrt {\langle \mathbf {M} ,\mathbf {M} \rangle }}={\sqrt {\operatorname {tr} \left(\mathbf {M} ^{*}\mathbf {M} \right)}}.}
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is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold, light blue bold in dark mode). Furthermore, because the matrices
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Mademlis, Ioannis; Tefas, Anastasios; Pitas, Ioannis (2018). "Regularized SVD-Based Video Frame
Saliency for Unsupervised Activity Video Summarization".
14841:{\displaystyle \mathbf {V} ^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} ={\bar {\mathbf {D} }}={\begin{bmatrix}\mathbf {D} &0\\0&0\end{bmatrix}},}
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This is an important property for applications for which invariance to the choice of units on variables (e.g., metric versus imperial units) is needed.
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to predict people's item ratings. Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines.
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29426:"Integrative Analysis of Genome-Scale Data by Using Pseudoinverse Projection Predicts Novel Correlation Between DNA Replication and RNA Transcription"
13966:{\displaystyle \nabla \mathbf {u} ^{\operatorname {T} }\mathbf {M} \mathbf {u} -\lambda \cdot \nabla \mathbf {u} ^{\operatorname {T} }\mathbf {u} =0}
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by the same unit-phase factor. In general, the SVD is unique up to arbitrary unitary transformations applied uniformly to the column vectors of both
29544:"SVD Identifies Transcript Length Distribution Functions from DNA Microarray Data and Reveals Evolutionary Forces Globally Affecting GBM Metabolism"
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16739:
27054:
This is an important property for applications in which it is necessary to preserve
Euclidean distances and invariance with respect to rotations.
8969:
31974:
27222:
21163:
is the Givens rotation matrix with the angle chosen such that the given pair of off-diagonal elements become equal after the rotation, and where
12799:
10446:
10029:
31441:
27832:
13226:
638:
20269:
14470:
31268:
18034:{\displaystyle \mathbf {\Sigma } ={\begin{bmatrix}{\begin{bmatrix}\mathbf {D} ^{\frac {1}{2}}&0\\0&0\end{bmatrix}}\\0\end{bmatrix}},}
7570:{\displaystyle {\begin{aligned}\mathbf {Mv} &=\sigma \mathbf {u} ,\\\mathbf {M} ^{*}\mathbf {u} &=\sigma \mathbf {v} .\end{aligned}}}
29929:
Hadi Fanaee Tork; João Gama (May 2015). "EigenEvent: An
Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance".
29163:
22405:
9798:
9603:
31996:
30162:{\displaystyle \operatorname {Tr} (\mathbf {V} _{2}^{*}\mathbf {M} ^{*}\mathbf {M} \mathbf {V} _{2})=\|\mathbf {M} \mathbf {V} _{2}\|^{2}}
29974:
Muralidharan, Vivek; Howell, Kathleen (2023). "Stretching directions in cislunar space: Applications for departures and transfer design".
28761:
in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:
18697:
11964:
The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix
7872:
are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that:
5588:
31210:
28856:
discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a
16936:
5038:
To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere
30434:
30285:. By Bai, Zhaojun; Demmel, James; Dongarra, Jack J.; Ruhe, Axel; van der Vorst, Henk A. Society for Industrial and Applied Mathematics.
22031:
subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small (
13353:
12419:
in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example,
31431:
29127:
22336:
22129:
There is an alternative way that does not explicitly use the eigenvalue decomposition. Usually the singular value problem of a matrix
31979:
31752:
31235:
29485:"Singular Value Decomposition of Genome-Scale mRNA Lengths Distribution Reveals Asymmetry in RNA Gel Electrophoresis Band Broadening"
13477:
32001:
21065:—is an iterative algorithm where a square matrix is iteratively transformed into a diagonal matrix. If the matrix is not square the
14201:
13558:
13480:
also has been applied for real time event detection from complex data streams (multivariate data with space and time dimensions) in
13409:. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be
11999:
5850:
31558:
31413:
29116:
29066:
20660:
32326:
31989:
31389:
22119:
16694:
8645:
Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor
972:
32219:
31230:
31082:
30017:
Muralidharan, Vivek; Howell, Kathleen (2022). "Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits".
16013:
12896:
12850:
11002:
10080:
4263:
4148:
918:
13491:, the SVD and its variants are used as an option to determine suitable maneuver directions for transfer trajectory design and
12730:
12551:
which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.
32392:
32072:
31870:
31047:
30956:
30937:
30914:
30889:
30821:
30713:
30640:
30298:
29334:
17132:
16798:
30223:
Rijk, P.P.M. de (1989). "A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer".
26063:
9987:
288:
32067:
28117:
in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If
27091:
26936:
26120:
10636:
1359:
26002:
21102:
22508:
20219:
29195:
DeAngelis, G. C.; Ohzawa, I.; Freeman, R. D. (October 1995). "Receptive-field dynamics in the central visual pathways".
20476:
19613:
11952:, as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see
5449:
31281:
30622:
29096:
29091:
28014:
27647:
26437:{\displaystyle \langle \mathbf {M} ,\mathbf {N} \rangle =\operatorname {tr} \left(\mathbf {N} ^{*}\mathbf {M} \right).}
10616:
2092:
30832:
8705:
are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of
1051:
with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as
32402:
32224:
31370:
31261:
31189:
31075:
29169:
28066:
17372:{\displaystyle {\bigl \{}\lambda _{i}^{-1/2}\mathbf {M} {\boldsymbol {v}}_{i}{\bigr \}}{\vphantom {|}}_{i=1}^{\ell }}
17089:{\displaystyle {\bigl \{}\lambda _{i}^{-1/2}\mathbf {M} {\boldsymbol {v}}_{i}{\bigr \}}{\vphantom {|}}_{i=1}^{\ell }}
12994:
9295:
The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently:
5411:
diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of
20181:
20141:
16885:
12220:{\displaystyle \mathbf {M} =\sum _{i}\mathbf {A} _{i}=\sum _{i}\sigma _{i}\mathbf {U} _{i}\otimes \mathbf {V} _{i}.}
8139:
A singular value for which we can find two left (or right) singular vectors that are linearly independent is called
227:
32042:
31640:
31220:
27695:
22200:
22160:
12689:
11872:
9514:
9403:
9334:
29684:"Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions"
29071:
28798:
28148:
27976:
26292:
21542:
21504:
21411:
21344:
18651:
18613:
18575:
18537:
17214:
16851:
15429:
15235:
15201:
15058:
14921:
14647:
9476:
3316:
32011:
31285:
31147:
10563:
10412:
1723:
30172:
29715:
Setyawati, Y.; Ohme, F.; Khan, S. (2019). "Enhancing gravitational waveform model through dynamic calibration".
15092:
12652:
32412:
31925:
31809:
29055:
29050:
22302:{\displaystyle {\begin{bmatrix}\mathbf {0} &\mathbf {M} \\\mathbf {M} ^{*}&\mathbf {0} \end{bmatrix}}.}
20848:
13451:, giving a handle on some of the uncertainty that should be allowed for around the current central prediction.
11516:
equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in
1215:
The SVD is not unique, however it is always possible to choose the decomposition such that the singular values
19116:
19073:
11180:
4461:{\displaystyle T:\left\{{\begin{aligned}K^{n}&\to K^{m}\\x&\mapsto \mathbf {M} x\end{aligned}}\right.}
684:
32234:
31745:
31436:
25053:
19064:{\displaystyle \sigma (\mathbf {u} ,\mathbf {v} )=\mathbf {u} ^{\operatorname {T} }\mathbf {M} \mathbf {v} ,}
13824:
12998:
11678:
11587:
10592:
25523:
25258:
24826:
24621:
24190:
24039:
19188:
32417:
31719:
31492:
31426:
31254:
31179:
29137:
28689:
28647:
26732:
26334:
22698:
21959:
and then use
Householder reflections to further reduce the matrix to bidiagonal form; the combined cost is
21186:
After the algorithm has converged the resulting diagonal matrix contains the singular values. The matrices
17440:
are in general not unitary, since they might not be square. However, we do know that the number of rows of
13439:
13319:
9584:
4669:
28909:. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by
25321:
24889:
24253:
8317:
As an exception, the left and right-singular vectors of singular value 0 comprise all unit vectors in the
3419:
except the geometric interpretation of the singular values as stretches is lost. In short, the columns of
32371:
32291:
31845:
31456:
30263:
29606:
29591:
28986:
24096:
22081:
21229:
21062:
19853:
Similar to the eigenvalues case, by assumption the two vectors satisfy the
Lagrange multiplier equation:
14955:
13472:
Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease
11209:
10628:
8809:
spanning the subspaces of each singular value, and up to arbitrary unitary transformations on vectors of
4640:
2847:
439:
29876:
Hadi Fanaee Tork; João Gama (September 2014). "Eigenspace method for spatiotemporal hotspot detection".
23617:
Visualization of
Reduced SVD variants. From top to bottom: 1: Full SVD, 2: Thin SVD (remove columns of
21476:
18086:
16354:
where the subscripts on the identity matrices are used to remark that they are of different dimensions.
13416:
13398:. It often controls the error rate or convergence rate of a given computational scheme on such systems.
12044:
11903:
11519:
10262:
9446:
8267:
7725:
6496:
2412:
2232:
1814:
1406:
1248:
727:
394:
32346:
32244:
32124:
31701:
31655:
31579:
31461:
31133:
30276:
30019:
29101:
29002:
28857:
28848:
could be made equal to another by independent orthogonal transformations of the two spaces it acts on.
25016:
non-zero singular values. The truncated SVD is no longer an exact decomposition of the original matrix
20604:
19546:
18052:
14875:
9947:
5351:
5131:
5067:
3452:
2948:
2811:
2683:
2129:
1465:
1146:
25466:
25193:
24761:
24680:
24588:
24125:
23927:
23894:
21825:
20571:
19760:
19727:
19689:
19656:
19513:
18978:
17910:
17877:
17776:
17747:
17443:
17414:
17385:
17248:
17185:
17103:
16106:
16077:
15400:
15371:
14353:
12891:
amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix
12355:
12267:
12234:
11427:
11048:
9914:
8181:
8148:
6615:
6548:
5318:
5261:
4767:
3939:
3853:
3516:
3362:
3283:
3216:
2915:
2778:
2745:
2650:
2524:
2372:
2329:
2262:
2191:
1993:
497:
32351:
32214:
32047:
32032:
31840:
31804:
31696:
31512:
31184:
29040:
28254:
27732:
26902:
are uniquely defined and are invariant with respect to left and/or right unitary transformations of
25588:
24713:
24326:
4717:
2362:
29862:
29805:
27596:
21964:
21268:
20735:
13190:
31943:
31933:
31814:
31738:
31548:
31446:
31349:
31098:
30984:
30967:
30744:
29086:
29081:
29045:
28877:
28731:
28619:
27390:
27361:
27329:
27185:
27156:
27062:
27030:
27001:
26907:
26703:
26660:
26182:
25752:
25650:
25596:
25179:{\displaystyle {\tilde {\mathbf {M} }}=\mathbf {U} _{t}\mathbf {\Sigma } _{t}\mathbf {V} _{t}^{*},}
25021:
24378:
24010:
21688:
The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an
18771:
14415:
14272:
13343:
13058:
12619:
11773:
11460:
11394:
11287:
11110:
11081:
10838:
10701:
10340:
10129:
9849:
9844:
is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase
9759:
8928:
8866:
8650:
8355:
8330:
8112:
8024:
7751:
3638:
3487:
3424:
3395:
2060:
1510:
1324:
1189:
28772:
28122:
27744:
27276:
27130:
26975:
26881:
25906:
25787:
25695:
25625:
25416:
24654:
24538:
24415:
23984:
23848:
23751:
23693:
22735:
22379:
22134:
21934:
21584:
21452:
21385:
21311:
20545:
20519:
19819:
19793:
19416:
19353:
18855:
18826:
18800:
18745:
17472:
16055:
15463:
15282:
15153:
14996:
14853:
14715:
14595:
14560:
14444:
14327:
14301:
14246:
14064:
14036:
13873:
13800:
13633:
13603:
13532:
13032:
13006:
12773:
12593:
12564:
12454:
12428:
12104:
11969:
11860:{\displaystyle {\tilde {\mathbf {M} }}=\mathbf {U} {\tilde {\mathbf {\Sigma } }}\mathbf {V} ^{*},}
11652:
11563:
11495:
11368:
11342:
11316:
11245:
11150:
10976:
10949:
10919:
10893:
10867:
10808:
10782:
10756:
10730:
10675:
10420:
10314:
10288:
10210:
10184:
9737:
9680:
9654:
9558:
9377:
9304:
8943:
8840:
8814:
8788:
8762:
8736:
8710:
8684:
8621:
8595:
8515:
8426:
8241:
8215:
8055:
7967:
7911:
7851:
7825:
7610:
7584:
7433:
7376:
7350:
7010:
6589:
6522:
5526:
5390:
4317:
4202:
4122:
4096:
4061:
4035:
3976:
3913:
3823:
3764:
3734:
3667:
3612:
3586:
3553:
3257:
3190:
2910:
coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn
2719:
2561:
2498:
2472:
2442:
2295:
2165:
1967:
1907:
1596:
1539:
1298:
1272:
1218:
889:
863:
837:
811:
781:
751:
610:
584:
558:
532:
447:
336:
264:
32397:
32306:
32281:
32099:
32088:
31799:
31645:
31421:
29784:
29122:
29035:
29029:
28943:
28861:
28588:
27565:
26629:
22759:
22686:{\displaystyle \mathbf {M} =z_{0}\mathbf {I} +z_{1}\sigma _{1}+z_{2}\sigma _{2}+z_{3}\sigma _{3}}
19582:
19482:
13492:
13323:
13315:
12420:
12388:
12324:
9883:
9767:
9706:
8322:
7780:
4762:
The geometric content of the SVD theorem can thus be summarized as follows: for every linear map
4585:
2876:
1027:
27962:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {V} ^{*}\cdot \mathbf {V} T_{f}\mathbf {V} ^{*}}
25359:
25226:
24927:
24794:
24291:
24158:
17862:{\displaystyle \mathbf {U} ={\begin{bmatrix}\mathbf {U} _{1}&\mathbf {U} _{2}\end{bmatrix}}}
15361:{\displaystyle \mathbf {V} ={\begin{bmatrix}\mathbf {V} _{1}&\mathbf {V} _{2}\end{bmatrix}}}
15175:
2055:". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as
1432:
32407:
32157:
32147:
32142:
31850:
31676:
31620:
31584:
30979:
30739:
29106:
28918:
28865:
28217:
27419:
26859:
In addition, the
Frobenius norm and the trace norm (the nuclear norm) are special cases of the
26342:
25876:
25291:
24859:
24223:
23663:
22559:
22089:
22055:
22039:
21817:
21771:
21617:
20933:
18881:
18108:
14687:
14619:
13980:
13659:
13455:
11948:
11946:
largest singular values (the other singular values are replaced by zero). This is known as the
11555:
10612:
10373:
8902:
7881:
5671:
3088:
3027:
2587:
2026:
1933:
1645:
1620:
1590:
1565:
471:
416:
362:
236:
201:
28098:
21660:
19442:
14391:
13508:
8363:
8293:
7644:
2617:
31902:
31383:
31143:
29849:
29792:
29273:
The
Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression
29143:
29019:
28841:
27736:
25556:
21605:
is typically computed by a two-step procedure. In the first step, the matrix is reduced to a
21094:
20715:
19379:
19329:
19284:
19247:
19164:
18911:
14007:
13792:
13406:
12475:
represents the time modulation (or vice versa). One may then define an index of separability
10551:{\displaystyle \mathbf {M} ^{+}=\mathbf {V} {\boldsymbol {\Sigma }}^{+}\mathbf {U} ^{\ast },}
9591:
8932:
8482:
8107:
with a subset of basis vectors spanning the right-singular vectors of each singular value of
7320:
4571:{\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),}
4377:
2224:
2085:
1889:
31379:
30373:"Maximum properties and inequalities for the eigenvalues of completely continuous operators"
25844:
24991:
23641:), 4: Truncated SVD (keep only largest t singular values and corresponding columns/rows in
21089:
matrix. The elementary iteration zeroes a pair of off-diagonal elements by first applying a
16426:{\displaystyle \mathbf {U} _{1}=\mathbf {M} \mathbf {V} _{1}\mathbf {D} ^{-{\frac {1}{2}}}.}
8897:
The singular value decomposition is very general in the sense that it can be applied to any
8396:
8019:
with a subset of basis vectors spanning the left-singular vectors of each singular value of
5313:
onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry
4982:
1134:{\displaystyle \mathbf {M} =\sum _{i=1}^{r}\sigma _{i}\mathbf {u} _{i}\mathbf {v} _{i}^{*},}
32316:
32295:
32209:
32094:
32057:
31659:
31138:
30777:
30770:
Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis
30587:
30532:
30384:
30028:
29983:
29895:
29837:
29734:
29621:
29555:
29496:
29437:
29378:
29111:
28758:
28291:
28186:
27801:
27770:
27538:
25813:
21696:). However, in practice it suffices to compute the SVD up to a certain precision, like the
21693:
20816:
18946:
17773:
are orthonormal and can be extended to an orthonormal basis. This means that we can choose
14901:
13865:
13481:
13448:
13410:
8679:(for the real case up to a sign). Consequently, if all singular values of a square matrix
8210:
right-singular vectors coincides, and these singular vectors appear in the same columns of
8081:
7993:
7459:
7402:
5552:
5490:
4927:
4848:
4817:
4343:
4232:
2358:
1713:{\displaystyle \mathbf {U} ^{*}\mathbf {U} =\mathbf {V} ^{*}\mathbf {V} =\mathbf {I} _{r}.}
185:
174:
31246:
31062:
30443:
21881:
flops, assuming that only the singular values are needed and not the singular vectors. If
21738:
21705:
21033:, and the singular values are given as the norms of the columns of the transformed matrix
14465:
gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there
10124:
Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of
5416:
5228:
5223:
can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid
5171:
188:
into a rotation, followed by a rescaling followed by another rotation. It generalizes the
8:
32119:
31855:
31625:
31563:
31277:
31117:
30809:
29174:
29132:
28978:
28906:
27892:
13466:
13402:
13331:
11142:
10236:
10158:
9979:
8541:
8452:
4005:
3793:
3245:
2220:
524:
30781:
30436:
A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
30388:
30032:
29987:
29899:
29841:
29738:
29625:
29559:
29500:
29441:
29382:
29009:
published a variant of the Golub/Kahan algorithm that is still the one most-used today.
28561:
28315:
27642:
This can be shown by mimicking the linear algebraic argument for the matrix case above.
27302:
26615:{\displaystyle \|\mathbf {M} \|={\sqrt {{\vphantom {\bigg |}}\sum _{i}\sigma _{i}^{2}}}}
26333:
The singular values are related to another norm on the space of operators. Consider the
26280:{\displaystyle \|\mathbf {M} \|=\operatorname {Tr} (\mathbf {M} ^{*}\mathbf {M} )^{1/2}}
24500:{\displaystyle \mathbf {M} =\mathbf {U} _{r}\mathbf {\Sigma } _{r}\mathbf {V} _{r}^{*}.}
23836:{\displaystyle \mathbf {M} =\mathbf {U} _{k}\mathbf {\Sigma } _{k}\mathbf {V} _{k}^{*},}
23613:
14530:
5637:
1858:
32249:
32178:
32109:
31953:
31915:
31650:
31517:
31005:
30901:
30868:
30844:
30793:
30701:
30689:
30591:
30520:
30482:
30407:
30372:
30353:
30044:
29999:
29956:
29938:
29911:
29885:
29827:
29750:
29724:
29578:
29543:
29519:
29484:
29340:
29272:
29220:
28901:
The fourth mathematician to discover the singular value decomposition independently is
28074:
27571:
27458:
25728:
25671:
25499:
25442:
25392:
25088:
24967:
24564:
24514:
24072:
23960:
23719:
23629:), 3: Compact SVD (remove vanishing singular values and corresponding columns/rows in
21910:
21886:
21209:
21189:
21166:
21146:
21072:
21036:
21016:
21010:
20992:
20972:
20910:
20796:
20776:
17514:
17494:
15133:
15038:
15018:
13278:
12725:
This intuitively makes sense because an orthogonal matrix would have the decomposition
12300:
11927:
11624:
8571:
7937:
5294:
5204:
5103:
5043:
5035:
is therefore represented by a diagonal matrix with non-negative real diagonal entries.
5016:
4958:
4903:
4879:
4616:
4470:
has a particularly simple description with respect to these orthonormal bases: we have
3889:
3710:
3693:
3357:
By the definition of a unitary matrix, the same is true for their conjugate transposes
3146:
3118:
3060:
2999:
2403:
189:
29667:
29650:
29634:
29460:
29425:
29367:"Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling"
29318:
2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
15460:
corresponding to non-zero and zero eigenvalues, respectively. Using this rewriting of
14584:
This section gives these two arguments for existence of singular value decomposition.
13223:
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix
8731:
by a unit-phase factor and simultaneous multiplication of the corresponding column of
32356:
32331:
32016:
31938:
31630:
31043:
30952:
30933:
30926:
30910:
30885:
30817:
30709:
30681:
30636:
30618:
30595:
30563:
30412:
30345:
30294:
30048:
30003:
29769:
29754:
29583:
29524:
29465:
29406:
29401:
29366:
29330:
29255:
29212:
29208:
29148:
29006:
28933:
28181:. Applying the diagonalization result, the unitary image of its positive square root
25646:
are of interest, which are more challenging to compute compared to the largest ones.
25551:
are calculated. This can be much quicker and more economical than the compact SVD if
21606:
13443:
13327:
13291:
12586:
11274:
Another application of the SVD is that it provides an explicit representation of the
10888:
can be characterized as a right-singular vector corresponding to a singular value of
9588:
6644:
4224:
3545:
2052:
1743:
1727:
911:
632:
30872:
30693:
30486:
30357:
29960:
29915:
29344:
29224:
32361:
32062:
31910:
31865:
31789:
31635:
31553:
31522:
31502:
31487:
31482:
31477:
31153:
30997:
30989:
30860:
30785:
30749:
30671:
30663:
30575:
30512:
30474:
30402:
30392:
30335:
30327:
30286:
30280:
30036:
29991:
29948:
29903:
29742:
29695:
29662:
29629:
29573:
29563:
29514:
29504:
29455:
29445:
29396:
29386:
29322:
29297:
29247:
29204:
29061:
28922:
28849:
27765:
need not be unitary is that, unlike the finite-dimensional case, given an isometry
27354:
27269:
24370:
22542:
22538:
22371:
22328:
22324:
22320:
22316:
21956:
21689:
21302:
The singular value decomposition can be computed using the following observations:
21066:
19321:
14679:
13684:
13625:
13295:
13274:
11487:
10365:
7960:
7310:
5011:
and sends the leftover basis vectors to zero. With respect to these bases, the map
3578:
803:
31314:
31035:
30442:, SIAM Journal on Matrix Analysis, vol. 239, pp. 781–800, archived from
22050:). This method computes the SVD of the bidiagonal matrix by solving a sequence of
32336:
32321:
32229:
32192:
32188:
32152:
32114:
32052:
32037:
32006:
31948:
31907:
31894:
31819:
31761:
31730:
31497:
31451:
31399:
31394:
31365:
31194:
30652:"Software suite for gene and protein annotation prediction and similarity search"
30583:
30528:
30500:
30430:
30316:"Software suite for gene and protein annotation prediction and similarity search"
29777:
29568:
28990:
28982:
28065:
The notion of singular values and left/right-singular vectors can be extended to
22494:{\displaystyle \mathbf {M} \Rightarrow \mathbf {Q} \mathbf {L} \mathbf {P} ^{*},}
21697:
21090:
20652:
13311:
11761:{\displaystyle \operatorname {rank} {\bigl (}{\tilde {\mathbf {M} }}{\bigr )}=r,}
10830:
6581:
5199:
and all the singular values are distinct and non-zero, the SVD of the linear map
5163:
3575:
3019:
2987:
1800:
1751:
1731:
31324:
31022:
30463:; Young, G. (1936). "The approximation of one matrix by another of lower rank".
29820:
29326:
28937:
28902:
25988:{\displaystyle \|\mathbf {M} \|=\|\mathbf {M} ^{*}\mathbf {M} \|^{\frac {1}{2}}}
11172:
10335:
are unitary matrices that are not necessarily related except through the matrix
32286:
32265:
32183:
32173:
31984:
31891:
31824:
31784:
31686:
31538:
31339:
31112:
30377:
Proceedings of the National Academy of Sciences of the United States of America
30252:
30241:
29746:
29542:
Bertagnolli, N. M.; Drake, J. A.; Tennessen, J. M.; Alter, O. (November 2013).
29316:
29153:
29024:
28929:
28853:
26686:
25873:
In other words, the Ky Fan 1-norm is the operator norm induced by the standard
22784:
21610:
13335:
12983:{\displaystyle \mathbf {M} =\mathbf {R} \mathbf {P} =\mathbf {P} '\mathbf {R} }
12644:
11644:
11543:
11538:. In numerical linear algebra the singular values can be used to determine the
11363:
and the left-singular vectors corresponding to the non-zero singular values of
7341:
6577:
4370:
3689:
2254:
1862:
1809:
1780:
1762:
773:
388:
182:
162:
21:
30667:
30633:
Analysis and Linear Algebra: The Singular Value Decomposition and Applications
30331:
30040:
29995:
29778:"Application of Dimensionality Reduction in Recommender System – A Case Study"
29700:
29683:
29301:
29251:
32386:
31691:
31615:
31344:
31329:
31319:
31158:
31018:
30805:
30765:
30761:
30727:
30559:
30465:
29773:
29391:
29285:
28998:
28994:
28914:
28845:
26860:
26057:
25779:
24708:
are not calculated. This is quicker and more economical than the thin SVD if
19317:
16788:{\displaystyle \mathbf {M} \mathbf {V} _{1}\mathbf {V} _{1}^{*}=\mathbf {M} }
13488:
11991:
10941:
9580:
3249:
3052:
1959:
1767:
1747:
193:
30651:
30503:(1958). "Inversion of Matrices by Biorthogonalization and Related Results".
30315:
30290:
29509:
29450:
18523:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}.}
12542:{\displaystyle \alpha ={\frac {\sigma _{1}^{2}}{\sum _{i}\sigma _{i}^{2}}},}
10615:
and transposing the resulting matrix. The pseudoinverse is one way to solve
9009:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*},}
32104:
31958:
31899:
31681:
31334:
31304:
31067:
30723:
30685:
30416:
30397:
30349:
29587:
29528:
29469:
29410:
29259:
28583:
Notice how this resembles the expression from the finite-dimensional case.
27259:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}}
22312:
22020:
21336:
19348:
is continuous, it attains a largest value for at least one pair of vectors
12836:{\displaystyle \mathbf {A} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}}
12416:
11275:
10483:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}}
10066:{\displaystyle \mathbf {S} =\mathbf {U} \mathbf {\Sigma } \mathbf {U} ^{*}}
9326:
7711:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}}
5383:
3756:
1893:
Visualization of the matrix multiplications in singular value decomposition
30851:
Hansen, P. C. (1987). "The truncated SVD as a method for regularization".
30768:(1965). "Calculating the singular values and pseudo-inverse of a matrix".
29216:
19651:
it must be non-negative. If it were negative, changing the sign of either
13258:{\displaystyle \mathbf {M} =\mathbf {A} ^{\operatorname {T} }\mathbf {B} }
672:{\displaystyle \mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{\mathrm {T} }.}
32301:
31886:
31610:
31600:
31507:
31309:
30460:
29158:
28910:
26209:
25717:
21496:
21333:
17100:
vectors. This matches with the matrix formalism used above denoting with
11311:
The right-singular vectors corresponding to vanishing singular values of
9595:
9468:
2981:
1735:
178:
30676:
30340:
28726:) can be considered the left-singular (resp. right-singular) vectors of
14516:{\displaystyle f(\mathbf {x} )=\mathbf {x} ^{*}\mathbf {M} \mathbf {x} }
13314:
and is useful in the analysis of regularization methods such as that of
9978:
The natural connection of the SVD to non-normal matrices is through the
1734:
of a matrix. The SVD is also extremely useful in all areas of science,
31794:
31543:
31375:
31009:
30864:
30797:
30579:
30524:
30478:
29952:
29907:
27891:
As for matrices, the singular value factorization is equivalent to the
22042:(GSL). The GSL also offers an alternative method that uses a one-sided
21766:
flops. Thus, the first step is more expensive, and the overall cost is
21700:. If this precision is considered constant, then the second step takes
13339:
11279:
8566:
orthogonal vectors from the kernel. However, if the singular value of
3845:
1739:
26850:{\displaystyle {\sqrt {{\vphantom {\bigg |}}\sum _{ij}|m_{ij}|^{2}}}.}
22537:
and repeat the orthogonalizations. Eventually, this iteration between
13458:
to interpolate solutions to three-dimensional unsteady flow problems.
7005:
This particular singular value decomposition is not unique. Choosing
31779:
31765:
31174:
30617:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
29364:
26208:
The last of the Ky Fan norms, the sum of all singular values, is the
25778:
The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the
24402:
which can make for a significantly quicker calculation in this case.
22440:{\displaystyle \mathbf {R} \Rightarrow \mathbf {L} \mathbf {P} ^{*}.}
22155:
is converted into an equivalent symmetric eigenvalue problem such as
20471:
More singular vectors and singular values can be found by maximizing
9835:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {D} \mathbf {U} ^{*}}
9640:{\displaystyle \mathbf {M} =\mathbf {U} \mathbf {D} \mathbf {U} ^{*}}
5123:
3968:
3138:
3080:
1776:
31001:
30993:
30789:
30753:
30566:(1970). "Singular value decomposition and least squares solutions".
30516:
20930:
After the algorithm has converged, the singular value decomposition
20638:
The passage from real to complex is similar to the eigenvalue case.
18731:{\displaystyle \mathbf {u} ^{\mathrm {T} }\mathbf {M} \mathbf {v} ,}
10611:, which is formed by replacing every non-zero diagonal entry by its
5623:{\displaystyle \mathbf {U} \circ \mathbf {D} \circ \mathbf {V} ^{*}}
2324:
Thus the SVD decomposition breaks down any linear transformation of
32366:
32311:
30847:. McGill University, CCGCR Report No. 97-1, Montréal, Québec, 52pp.
29729:
26547:
Since the trace is invariant under unitary equivalence, this shows
24321:
The thin SVD uses significantly less space and computation time if
22311:
The approaches that use eigenvalue decompositions are based on the
16984:{\displaystyle \{\mathbf {M} {\boldsymbol {v}}_{i}\}_{i=1}^{\ell }}
13473:
13299:
12449:
in the SVD factorization is then a Gabor while the first column of
8318:
29943:
29890:
29832:
25584:
but requires a completely different toolset of numerical solvers.
18795:
over particular subspaces. The singular vectors are the values of
17469:
is no smaller than the number of columns, since the dimensions of
13391:{\displaystyle \kappa :=\sigma _{\text{max}}/\sigma _{\text{min}}}
13342:
can be determined from the singular vectors. Yet another usage is
10944:
and has no vanishing singular value, the equation has no non-zero
30898:
30879:
30656:
IEEE/ACM Transactions on Computational Biology and Bioinformatics
30320:
IEEE/ACM Transactions on Computational Biology and Bioinformatics
22023:
for the computation of eigenvalues, which was first described by
2991:
1790:
31033:
30708:. Philadelphia: Society for Industrial and Applied Mathematics.
22361:{\displaystyle \mathbf {M} \Rightarrow \mathbf {Q} \mathbf {R} }
20773:
is chosen such that after the rotation the columns with numbers
16736:
This can be also seen as immediate consequence of the fact that
11643:. In the case that the approximation is based on minimizing the
9467:(non-zero singular values) are the square roots of the non-zero
7311:
Singular values, singular vectors, and their relation to the SVD
31225:
31215:
31034:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),
31023:"Singular value decomposition and principal component analysis"
25721:
22028:
18692:
The singular values can also be characterized as the maxima of
18047:
to make the number of zero rows equal the number of columns of
14439:
The same calculation performed on the orthogonal complement of
10411:
The singular value decomposition can be used for computing the
3172:
1897:
884:
are called left-singular vectors and right-singular vectors of
553:. Such decomposition always exists for any complex matrix. If
29541:
29286:"Local spectral variability features for speaker verification"
27524:{\displaystyle \mathbf {M} =\mathbf {U} T_{f}\mathbf {V} ^{*}}
14232:{\displaystyle \mathbf {M} \mathbf {u} =\lambda \mathbf {u} ,}
13589:{\displaystyle \mathbf {M} \mathbf {u} =\lambda \mathbf {u} .}
12030:{\displaystyle \mathbf {A} =\mathbf {u} \otimes \mathbf {v} ,}
6580:, multiplying by their respective conjugate transposes yields
5879:{\displaystyle \mathbf {U} \mathbf {\Sigma } \mathbf {V} ^{*}}
30505:
Journal of the Society for Industrial and Applied Mathematics
29076:
28840:
The singular value decomposition was originally developed by
12990:
in either order of stretch and rotation, as described above.
3699:
25808:
as a linear operator with respect to the Euclidean norms of
21069:
is performed first and then the algorithm is applied to the
20989:
is the accumulation of Jacobi rotation matrices, the matrix
13290:
The SVD and pseudoinverse have been successfully applied to
10231:
is not necessarily positive semi-definite, while the SVD is
9793:
will be non-negative real numbers so that the decomposition
3113:
matrix can be viewed as the magnitude of the semiaxis of an
1534:
and has only the non-zero singular values. In this variant,
31276:
30880:
Horn, Roger A.; Johnson, Charles R. (1985). "Section 7.3".
30730:(1990). "Accurate singular values of bidiagonal matrices".
30282:
Templates for the Solution of Algebraic Eigenvalue Problems
30264:
mathworks.co.kr/matlabcentral/fileexchange/12674-simple-svd
17382:
We see that this is almost the desired result, except that
13776:
13310:
The SVD is also applied extensively to the study of linear
9398:(referred to as left-singular vectors) are eigenvectors of
5256:
and specifically its axes; then consider the directions in
4455:
1722:
Mathematical applications of the SVD include computing the
30968:"On the Early History of the Singular Value Decomposition"
30928:
Foundations of Multidimensional and Metric Data Structures
29967:
27879:{\displaystyle {\begin{bmatrix}U_{1}\\U_{2}\end{bmatrix}}}
22118:, §8.6.3). Yet another method for step 2 uses the idea of
16729:{\displaystyle \mathbf {M} \mathbf {V} _{2}=\mathbf {0} .}
13318:. It is widely used in statistics, where it is related to
11240:
The solution turns out to be the right-singular vector of
10777:
is to be determined which satisfies the equation. Such an
5845:
A singular value decomposition of this matrix is given by
5485:
then sends the unit-sphere onto an ellipsoid isometric to
3633:
are both equal to the unitary matrix used to diagonalize
2990:
can be interpreted as the magnitude of the semiaxes of an
1015:{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n},}
29928:
29875:
29767:
16045:{\displaystyle \mathbf {M} \mathbf {V} _{2}=\mathbf {0} }
14027:
12928:{\displaystyle \mathbf {R} =\mathbf {U} \mathbf {V} ^{*}}
12882:{\displaystyle \mathbf {A} =\mathbf {U} \mathbf {V} ^{*}}
11554:
Some practical applications need to solve the problem of
11034:{\displaystyle \mathbf {x} ^{*}\mathbf {A} =\mathbf {0} }
10112:{\displaystyle \mathbf {R} =\mathbf {U} \mathbf {V} ^{*}}
4303:{\displaystyle \mathbf {V} _{1},\ldots ,\mathbf {V} _{n}}
4188:{\displaystyle \mathbf {U} _{1},\ldots ,\mathbf {U} _{m}}
958:{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{m}}
802:. The number of non-zero singular values is equal to the
29365:
O. Alter, P. O. Brown and D. Botstein (September 2000).
29283:
29237:
28757:
Compact operators on a Hilbert space are the closure of
28060:
20330:
Plugging this into the pair of equations above, we have
20321:{\displaystyle \sigma _{1}=2\lambda _{1}=2\lambda _{2}.}
18687:
13628:, a variational characterization is also available. Let
12759:{\displaystyle \mathbf {U} \mathbf {I} \mathbf {V} ^{*}}
21093:
to symmetrize the pair of elements and then applying a
21061:
Two-sided Jacobi SVD algorithm—a generalization of the
17175:{\displaystyle \{{\boldsymbol {v}}_{i}\}_{i=1}^{\ell }}
16841:{\displaystyle \{{\boldsymbol {v}}_{i}\}_{i=1}^{\ell }}
11455:
and the range is spanned by the first three columns of
10622:
10153:
while related, differ: the eigenvalue decomposition is
8892:
5162:
Non-zero singular values are simply the lengths of the
3244:
are unitary, the columns of each of them form a set of
2051:
matrices too. In that case, "unitary" is the same as "
31042:(3rd ed.), New York: Cambridge University Press,
29194:
27841:
27212:
26108:{\displaystyle \|\mathbf {M} ^{*}\mathbf {M} \|^{1/2}}
22250:
18354:
18312:
18261:
18195:
18187:
18145:
17968:
17960:
17822:
15826:
15615:
15568:
15496:
15321:
14802:
13438:
One application of SVD to rather large matrices is in
11768:
it turns out that the solution is given by the SVD of
10368:
square matrices have an eigenvalue decomposition, any
10015:{\displaystyle \mathbf {M} =\mathbf {S} \mathbf {R} ,}
8477:
orthogonal vectors from the cokernel. Conversely, if
7060:
6832:
6689:
6252:
6077:
5917:
5718:
2982:
Singular values as semiaxes of an ellipse or ellipsoid
2645:
it can be interpreted as a linear transformation from
324:{\displaystyle \mathbf {M} =\mathbf {U\Sigma V^{*}} ,}
31016:
30175:
30067:
29821:"Dimension Independent Matrix Square Using MapReduce"
28977:
Practical methods for computing the SVD date back to
28946:
28880:
28801:
28775:
28734:
28692:
28650:
28622:
28591:
28564:
28344:
28318:
28294:
28257:
28220:
28189:
28151:
28125:
28101:
28077:
28017:
27979:
27903:
27835:
27804:
27773:
27747:
27698:
27650:
27599:
27574:
27541:
27485:
27461:
27422:
27393:
27364:
27332:
27305:
27279:
27225:
27188:
27159:
27133:
27116:{\displaystyle \mathbf {D} \mathbf {A} \mathbf {E} ,}
27094:
27065:
27033:
27004:
26978:
26961:{\displaystyle \mathbf {U} \mathbf {A} \mathbf {V} ,}
26939:
26910:
26884:
26783:
26735:
26706:
26663:
26632:
26555:
26454:
26375:
26345:
26295:
26218:
26185:
26168:{\displaystyle (\mathbf {M} ^{*}\mathbf {M} )^{1/2},}
26123:
26066:
26005:
25935:
25909:
25879:
25847:
25816:
25790:
25755:
25731:
25698:
25674:
25628:
25599:
25587:
In applications that require an approximation to the
25559:
25526:
25502:
25469:
25445:
25419:
25395:
25362:
25324:
25294:
25261:
25229:
25196:
25114:
25091:
25056:
25024:
24994:
24970:
24930:
24892:
24862:
24829:
24797:
24764:
24716:
24683:
24657:
24624:
24591:
24567:
24541:
24517:
24444:
24418:
24381:
24369:
The first stage in its calculation will usually be a
24329:
24294:
24256:
24226:
24193:
24161:
24128:
24099:
24075:
24042:
24013:
23987:
23963:
23930:
23897:
23851:
23780:
23754:
23722:
23696:
23666:
22795:
22762:
22738:
22701:
22590:
22562:
22511:
22457:
22408:
22382:
22339:
22244:
22203:
22163:
22137:
22092:
22058:
21967:
21937:
21913:
21889:
21828:
21774:
21741:
21708:
21663:
21620:
21587:
21545:
21507:
21479:
21455:
21414:
21388:
21347:
21314:
21271:
21232:
21212:
21192:
21169:
21149:
21105:
21075:
21039:
21019:
20995:
20975:
20936:
20913:
20851:
20819:
20799:
20779:
20738:
20718:
20663:
20607:
20574:
20548:
20522:
20479:
20338:
20272:
20222:
20184:
20144:
19988:
19861:
19822:
19796:
19763:
19730:
19692:
19659:
19616:
19585:
19549:
19516:
19485:
19445:
19419:
19382:
19356:
19332:
19287:
19250:
19191:
19167:
19119:
19076:
19010:
18981:
18949:
18914:
18884:
18858:
18829:
18803:
18774:
18748:
18700:
18654:
18616:
18578:
18540:
18486:
18139:
18111:
18089:
18055:
17946:
17913:
17880:
17808:
17779:
17750:
17539:
17517:
17497:
17475:
17446:
17417:
17388:
17280:
17251:
17217:
17188:
17135:
17106:
16997:
16939:
16888:
16854:
16801:
16742:
16697:
16443:
16365:
16140:
16109:
16080:
16058:
16016:
15882:
15490:
15466:
15432:
15403:
15374:
15307:
15285:
15238:
15204:
15178:
15156:
15136:
15095:
15061:
15041:
15021:
14999:
14958:
14924:
14904:
14878:
14856:
14742:
14718:
14690:
14650:
14622:
14598:
14563:
14533:
14473:
14447:
14418:
14394:
14356:
14330:
14304:
14275:
14249:
14204:
14093:
14067:
14039:
14010:
13983:
13902:
13876:
13827:
13803:
13795:, this continuous function attains a maximum at some
13695:
13662:
13636:
13606:
13561:
13535:
13511:
13419:
13356:
13229:
13193:
13090:
13061:
13035:
13009:
12943:
12899:
12853:
12802:
12776:
12733:
12692:
12655:
12622:
12596:
12567:
12483:
12457:
12431:
12391:
12358:
12327:
12303:
12270:
12237:
12133:
12107:
12047:
12002:
11972:
11930:
11906:
11875:
11805:
11776:
11714:
11681:
11655:
11627:
11590:
11566:
11522:
11498:
11463:
11430:
11397:
11371:
11345:
11319:
11290:
11248:
11212:
11183:
11153:
11113:
11084:
11051:
11005:
10979:
10952:
10922:
10896:
10870:
10841:
10811:
10785:
10759:
10733:
10704:
10678:
10661:{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {0} }
10639:
10595:
10566:
10500:
10449:
10423:
10376:
10343:
10317:
10291:
10265:
10239:
10213:
10187:
10161:
10132:
10083:
10032:
9990:
9950:
9917:
9886:
9852:
9801:
9770:
9740:
9709:
9683:
9657:
9606:
9561:
9517:
9479:
9449:
9406:
9380:
9337:
9307:
9026:
8972:
8946:
8905:
8869:
8843:
8817:
8791:
8765:
8739:
8713:
8687:
8653:
8624:
8598:
8574:
8544:
8518:
8485:
8455:
8429:
8399:
8366:
8333:
8296:
8270:
8244:
8218:
8184:
8151:
8115:
8084:
8058:
8027:
7996:
7970:
7940:
7914:
7884:
7854:
7828:
7783:
7754:
7728:
7677:
7647:
7613:
7587:
7493:
7462:
7436:
7405:
7379:
7353:
7323:
7039:
7013:
6655:
6618:
6592:
6551:
6525:
6499:
5895:
5853:
5704:
5674:
5640:
5591:
5555:
5529:
5521:
To define the third and last move, apply an isometry
5493:
5452:
5419:
5393:
5354:
5321:
5297:
5264:
5231:
5207:
5174:
5134:
5106:
5070:
5046:
5019:
4985:
4961:
4930:
4906:
4882:
4851:
4820:
4770:
4720:
4672:
4643:
4619:
4588:
4478:
4387:
4346:
4320:
4266:
4235:
4205:
4151:
4125:
4099:
4064:
4038:
4008:
3979:
3942:
3916:
3892:
3856:
3826:
3796:
3767:
3737:
3713:
3688:
is not positive-semidefinite and Hermitian but still
3670:
3641:
3615:
3589:
3556:
3519:
3490:
3455:
3427:
3398:
3365:
3319:
3286:
3260:
3219:
3193:
3149:
3121:
3091:
3063:
3030:
3002:
2951:
2918:
2879:
2850:
2814:
2781:
2748:
2722:
2686:
2653:
2620:
2590:
2564:
2527:
2501:
2475:
2445:
2415:
2375:
2332:
2298:
2265:
2235:
2194:
2168:
2132:
2095:
2063:
2029:
1996:
1970:
1936:
1910:
1817:
1654:
1623:
1599:
1568:
1542:
1513:
1468:
1435:
1409:
1392:{\displaystyle \mathbf {M} =\mathbf {U\Sigma V} ^{*}}
1362:
1327:
1301:
1275:
1251:
1221:
1192:
1149:
1059:
1030:
975:
921:
892:
866:
840:
814:
784:
754:
730:
687:
641:
635:
matrices; in such contexts, the SVD is often denoted
613:
587:
561:
535:
500:
474:
450:
419:
397:
365:
339:
291:
267:
239:
233:
Specifically, the singular value decomposition of an
204:
30732:
SIAM Journal on Scientific and Statistical Computing
30635:. Student Mathematical Library (1st ed.). AMS.
29818:
29651:"On the singular values of Gaussian random matrices"
26727:
Direct calculation shows that the Frobenius norm of
26047:{\displaystyle (\mathbf {M} ^{*}\mathbf {M} )^{1/2}}
22584:
matrix can be found analytically. Let the matrix be
12993:
A similar problem, with interesting applications in
8935:. Nevertheless, the two decompositions are related.
3055:
being viewed as the magnitude of the semiaxis of an
30845:"A manual for EOF and SVD analyses of climate data"
29482:
29423:
28864:for bilinear forms under orthogonal substitutions.
28556:where the series converges in the norm topology on
22528:{\displaystyle \mathbf {M} \Leftarrow \mathbf {L} }
21929:then it is advantageous to first reduce the matrix
20257:{\displaystyle \|\mathbf {u} \|=\|\mathbf {v} \|=1}
18532:Notice the argument could begin with diagonalizing
11136:
8861:spanning the kernel and cokernel, respectively, of
5346:sending these directions to the coordinate axes of
1024:and if they are sorted so that the singular values
32342:Spectral theory of ordinary differential equations
31760:
31606:Spectral theory of ordinary differential equations
31040:Numerical Recipes: The Art of Scientific Computing
31025:. In D.P. Berrar; W. Dubitzky; M. Granzow (eds.).
30925:
30900:
30843:Halldor, Bjornsson and Venegas, Silvia A. (1997).
30205:
30161:
30016:
29973:
29681:
29607:"On the distribution of a scaled condition number"
29314:
28959:
28891:
28821:
28783:
28745:
28716:
28674:
28633:
28604:
28573:
28546:
28327:
28300:
28276:
28239:
28202:
28171:
28133:
28107:
28083:
28047:
27999:
27961:
27878:
27817:
27786:
27755:
27721:
27680:
27630:
27580:
27554:
27523:
27467:
27443:
27404:
27375:
27343:
27314:
27287:
27258:
27199:
27170:
27141:
27115:
27076:
27044:
27015:
26986:
26960:
26921:
26892:
26849:
26765:
26717:
26674:
26645:
26614:
26537:
26436:
26357:
26315:
26279:
26196:
26167:
26107:
26046:
25987:
25927:on (possibly infinite-dimensional) Hilbert spaces
25917:
25892:
25863:
25829:
25798:
25766:
25737:
25706:
25680:
25636:
25610:
25574:
25541:
25508:
25484:
25451:
25427:
25401:
25377:
25344:
25306:
25276:
25244:
25211:
25178:
25097:
25073:
25046:
25035:
25006:
24976:
24945:
24912:
24874:
24844:
24812:
24779:
24746:
24698:
24665:
24639:
24606:
24573:
24549:
24523:
24499:
24426:
24392:
24359:
24309:
24276:
24238:
24208:
24176:
24143:
24110:
24081:
24057:
24024:
23995:
23969:
23945:
23912:
23881:
23835:
23762:
23728:
23704:
23678:
23598:
22775:
22746:
22730:are complex numbers that parameterize the matrix,
22722:
22685:
22574:
22527:
22493:
22439:
22390:
22360:
22301:
22226:
22186:
22145:
22104:
22070:
22002:
21945:
21919:
21895:
21871:
21799:
21756:
21723:
21678:
21645:
21595:
21565:
21527:
21487:
21463:
21434:
21396:
21367:
21322:
21286:
21257:
21218:
21198:
21175:
21155:
21133:
21081:
21045:
21025:
21001:
20981:
20961:
20919:
20899:
20837:
20805:
20785:
20765:
20724:
20702:
20625:
20589:
20556:
20530:
20505:{\displaystyle \sigma (\mathbf {u} ,\mathbf {v} )}
20504:
20458:
20320:
20256:
20206:
20166:
20126:
19970:
19838:
19804:
19778:
19745:
19707:
19674:
19642:{\displaystyle \sigma (\mathbf {u} ,\mathbf {v} )}
19641:
19598:
19567:
19531:
19498:
19467:
19427:
19401:
19364:
19338:
19306:
19269:
19232:
19173:
19149:
19106:
19063:
18996:
18967:
18933:
18896:
18866:
18837:
18811:
18785:
18756:
18730:
18674:
18636:
18598:
18560:
18522:
18468:
18123:
18097:
18073:
18033:
17928:
17895:
17861:
17794:
17765:
17734:
17523:
17503:
17483:
17461:
17432:
17403:
17371:
17266:
17237:
17203:
17174:
17121:
17088:
16983:
16925:
16874:
16840:
16787:
16728:
16681:
16425:
16344:
16124:
16095:
16066:
16044:
16000:
15864:
15474:
15452:
15418:
15389:
15360:
15293:
15271:
15224:
15190:
15164:
15142:
15122:
15081:
15047:
15027:
15007:
14985:
14944:
14910:
14890:
14864:
14840:
14726:
14702:
14670:
14634:
14606:
14571:
14542:
14515:
14455:
14429:
14400:
14376:
14338:
14312:
14286:
14257:
14231:
14186:
14075:
14047:
14016:
13992:
13965:
13884:
13856:
13811:
13781:
13674:
13644:
13614:
13588:
13543:
13517:
13427:
13401:The SVD also plays a crucial role in the field of
13390:
13257:
13212:
13177:
13072:
13043:
13017:
12982:
12927:
12881:
12835:
12784:
12758:
12715:
12674:
12633:
12604:
12575:
12541:
12465:
12439:
12404:
12373:
12340:
12309:
12285:
12252:
12219:
12115:
12089:
12029:
11980:
11936:
11914:
11892:
11859:
11787:
11760:
11698:
11663:
11633:
11607:
11574:
11530:
11506:
11474:
11445:
11408:
11379:
11353:
11327:
11301:
11256:
11232:
11196:
11161:
11124:
11095:
11066:
11033:
10987:
10960:
10930:
10904:
10878:
10852:
10819:
10793:
10767:
10741:
10715:
10686:
10660:
10603:
10581:
10550:
10482:
10431:
10388:
10354:
10325:
10299:
10273:
10247:
10221:
10195:
10169:
10143:
10111:
10065:
10014:
9968:
9932:
9899:
9868:
9834:
9783:
9748:
9722:
9691:
9665:
9639:
9569:
9540:
9499:
9457:
9429:
9388:
9360:
9315:
9285:
9008:
8954:
8917:
8880:
8851:
8825:
8799:
8773:
8747:
8721:
8695:
8669:
8642:already appear as left or right-singular vectors.
8632:
8606:
8580:
8556:
8526:
8500:
8467:
8437:
8411:
8381:
8344:
8305:
8278:
8252:
8226:
8199:
8166:
8126:
8097:
8066:
8038:
8009:
7978:
7946:
7922:
7896:
7862:
7836:
7810:
7765:
7736:
7710:
7656:
7621:
7595:
7569:
7475:
7444:
7418:
7387:
7361:
7329:
7292:
7021:
6995:
6633:
6600:
6566:
6533:
6507:
6481:
5878:
5835:
5686:
5649:
5622:
5573:
5537:
5511:
5476:{\displaystyle \mathbf {D} \circ \mathbf {V} ^{*}}
5475:
5434:
5401:
5372:
5336:
5303:
5279:
5246:
5213:
5189:
5152:
5112:
5088:
5052:
5025:
5001:
4967:
4943:
4912:
4888:
4864:
4833:
4802:
4750:
4702:
4654:
4625:
4601:
4570:
4460:
4359:
4328:
4302:
4248:
4213:
4187:
4133:
4107:
4072:
4046:
4020:
3987:
3957:
3924:
3898:
3871:
3834:
3808:
3775:
3745:
3719:
3678:
3652:
3623:
3597:
3564:
3534:
3501:
3473:
3438:
3409:
3380:
3347:
3301:
3268:
3234:
3201:
3155:
3127:
3103:
3069:
3042:
3008:
2969:
2933:
2900:
2861:
2832:
2796:
2763:
2730:
2704:
2668:
2635:
2602:
2572:
2542:
2509:
2483:
2453:
2423:
2390:
2347:
2314:
2280:
2243:
2209:
2176:
2150:
2114:
2074:
2041:
2011:
1978:
1948:
1918:
1825:
1726:, matrix approximation, and determining the rank,
1712:
1635:
1607:
1580:
1550:
1524:
1495:
1450:
1417:
1391:
1338:
1309:
1283:
1259:
1237:
1203:
1176:
1133:
1043:
1014:
957:
900:
874:
848:
822:
792:
762:
738:
716:
671:
621:
595:
569:
543:
515:
486:
458:
431:
405:
377:
347:
323:
275:
251:
216:
32240:Schröder–Bernstein theorems for operator algebras
30615:Linear Algebra and Matrix Analysis for Statistics
29714:
28048:{\displaystyle \mathbf {V} T_{f}\mathbf {V} ^{*}}
27681:{\displaystyle \mathbf {V} T_{f}\mathbf {V} ^{*}}
26866:
26791:
26577:
17211:the matrix whose columns are the eigenvectors of
15586:
15585:
13001:, which consists of finding an orthogonal matrix
12559:It is possible to use the SVD of a square matrix
10833:and is sometimes called a (right) null vector of
7305:
2115:{\displaystyle \mathbf {x} \mapsto \mathbf {Ax} }
25:Illustration of the singular value decomposition
32384:
31027:A Practical Approach to Microarray Data Analysis
30700:
30010:
28993:. However, these were replaced by the method of
28917:in 1936; they saw it as a generalization of the
24723:
24336:
23858:
22123:
22019:The second step can be done by a variant of the
22013:
21810:
21655:floating-point operations (flop), assuming that
16795:. This is equivalent to the observation that if
14965:
14872:is diagonal and positive definite, of dimension
14678:is positive semi-definite and Hermitian, by the
14587:
11549:
7790:
4727:
4547:
2880:
1833:along the coordinate axes, and a final rotation
1475:
1156:
442:with non-negative real numbers on the diagonal,
30649:
30313:
28249:corresponding to strictly positive eigenvalues
26212:(also known as the 'nuclear norm'), defined by
22549:
12319:-th columns of the corresponding SVD matrices,
8421:then the cokernel is nontrivial, in which case
7371:if and only if there exist unit-length vectors
3696:and singular value decomposition are distinct.
1766:Animated illustration of the SVD of a 2D, real
29284:Sahidullah, Md.; Kinnunen, Tomi (March 2016).
28940:in 1910, who is the first to call the numbers
24616:corresponding to the non-zero singular values
20207:{\displaystyle \mathbf {v} _{1}^{\textrm {T}}}
20167:{\displaystyle \mathbf {u} _{1}^{\textrm {T}}}
16926:{\displaystyle \{\lambda _{i}\}_{i=1}^{\ell }}
11266:corresponding to the smallest singular value.
8050:It is always possible to find a unitary basis
7302:is also a valid singular value decomposition.
1799:into three simple transformations: an initial
31746:
31262:
31083:
30804:
29819:Bosagh Zadeh, Reza; Carlsson, Gunnar (2013).
27722:{\displaystyle \mathbf {M} ^{*}\mathbf {M} ,}
23068:
22943:
22227:{\displaystyle \mathbf {M} ^{*}\mathbf {M} ,}
22187:{\displaystyle \mathbf {M} \mathbf {M} ^{*},}
22115:
21056:
20646:
19717:would make it positive and therefore larger.
17333:
17283:
17050:
17000:
12716:{\displaystyle \mathbf {U} \mathbf {V} ^{*}.}
11893:{\displaystyle {\tilde {\mathbf {\Sigma } }}}
11744:
11723:
11422:the null space is spanned by the last row of
11269:
10914:that is zero. This observation means that if
10415:of a matrix. The pseudoinverse of the matrix
9541:{\displaystyle \mathbf {M} \mathbf {M} ^{*}.}
9430:{\displaystyle \mathbf {M} \mathbf {M} ^{*}.}
9361:{\displaystyle \mathbf {M} ^{*}\mathbf {M} .}
2773:can be chosen to be rotations/reflections of
31097:
30899:Horn, Roger A.; Johnson, Charles R. (1991).
30558:
30182:
30176:
30150:
30129:
29164:Two-dimensional singular-value decomposition
28822:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
28711:
28693:
28669:
28651:
28271:
28258:
28234:
28221:
28172:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
28000:{\displaystyle \mathbf {U} \mathbf {V} ^{*}}
26564:
26556:
26487:
26471:
26463:
26455:
26392:
26376:
26316:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
26227:
26219:
26088:
26067:
25971:
25950:
25944:
25936:
23746:The thin, or economy-sized, SVD of a matrix
22787:. Then its two singular values are given by
21566:{\displaystyle \mathbf {M} \mathbf {M} ^{*}}
21528:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
21435:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
21368:{\displaystyle \mathbf {M} \mathbf {M} ^{*}}
20703:{\displaystyle M\leftarrow MJ(p,q,\theta ),}
20245:
20237:
20231:
20223:
20136:Multiplying the first equation from left by
18675:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
18637:{\displaystyle \mathbf {M} \mathbf {M} ^{*}}
18599:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
18561:{\displaystyle \mathbf {M} \mathbf {M} ^{*}}
17238:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
17152:
17136:
16961:
16940:
16903:
16889:
16875:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
16818:
16802:
15453:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
15272:{\displaystyle {\bar {\mathbf {D} }}_{jj}=0}
15225:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
15082:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
14945:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
14671:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
13848:
13839:
13831:
13828:
13201:
13194:
13131:
13109:
12554:
11221:
11213:
10283:is diagonal and positive semi-definite, and
9500:{\displaystyle \mathbf {M} ^{*}\mathbf {M} }
9325:(referred to as right-singular vectors) are
8391:Even if all singular values are nonzero, if
5444:as stretching coefficients. The composition
3348:{\displaystyle \sigma _{i}\mathbf {U} _{i}.}
2895:
2883:
1898:Rotation, coordinate scaling, and reflection
1757:
1490:
1478:
1171:
1159:
30951:(3rd ed.). Wellesley-Cambridge Press.
30722:
30612:
30459:
29682:Walton, S.; Hassan, O.; Morgan, K. (2013).
29308:
22032:
16882:corresponding to non-vanishing eigenvalues
13553:is characterized by the algebraic relation
10582:{\displaystyle {\boldsymbol {\Sigma }}^{+}}
2435:followed by another rotation or reflection
54:, indicated by its effect on the unit disc
31753:
31739:
31269:
31255:
31090:
31076:
30760:
30546:
30206:{\displaystyle \|A\|=0\Leftrightarrow A=0}
29128:Non-linear iterative partial least squares
22024:
19508:and the corresponding vectors are denoted
15123:{\displaystyle {\bar {\mathbf {D} }}_{ii}}
14030:operator (differentiation with respect to
12675:{\displaystyle \mathbf {O} -\mathbf {A} .}
12350:are the ordered singular values, and each
10401:
10075:is positive semidefinite and normal, and
5583:As can be easily checked, the composition
3700:Relation to the four fundamental subspaces
2994:in 2D. This concept can be generalized to
31017:Wall, Michael E.; Rechtsteiner, Andreas;
30983:
30946:
30923:
30743:
30675:
30406:
30396:
30339:
30061:To see this, we just have to notice that
29942:
29889:
29831:
29728:
29699:
29666:
29633:
29577:
29567:
29518:
29508:
29459:
29449:
29400:
29390:
28932:defined an analog of singular values for
28844:, who wished to determine whether a real
26325:are the squares of the singular values).
24649:are calculated. The remaining vectors of
22716:
22038:The same algorithm is implemented in the
21406:are a set of orthonormal eigenvectors of
20900:{\displaystyle (p=1\dots m,q=p+1\dots m)}
18984:
17274:the matrix whose columns are the vectors
15562:
13730:
13712:
9203:
9084:
6584:, as shown below. In this case, because
31559:Group algebra of a locally compact group
30837:GNU Scientific Library. Reference Manual
30830:
30613:Banerjee, Sudipto; Roy, Anindya (2014),
30499:
29117:Multilinear principal component analysis
29067:Generalized singular value decomposition
23612:
22120:divide-and-conquer eigenvalue algorithms
22047:
19150:{\displaystyle \mathbf {v} \in S^{n-1}.}
19107:{\displaystyle \mathbf {u} \in S^{m-1},}
13465:Singular value decomposition is used in
12643:The closeness of fit is measured by the
11197:{\displaystyle \mathbf {A} \mathbf {x} }
3083:. Similarly, the singular values of any
1888:
1761:
717:{\displaystyle \sigma _{i}=\Sigma _{ii}}
20:
31221:Basic Linear Algebra Subprograms (BLAS)
31029:. Norwell, MA: Kluwer. pp. 91–109.
30965:
30630:
30429:
29604:
29424:O. Alter; G. H. Golub (November 2004).
27086:In other words, the singular values of
26931:In other words, the singular values of
26328:
25074:{\displaystyle {\tilde {\mathbf {M} }}}
21495:) are the square roots of the non-zero
19788:are left and right-singular vectors of
17321:
17141:
17038:
16950:
16807:
13857:{\displaystyle \{\|\mathbf {x} \|=1\}.}
13268:
11953:
11699:{\displaystyle {\tilde {\mathbf {M} }}}
11608:{\displaystyle {\tilde {\mathbf {M} }}}
10604:{\displaystyle {\boldsymbol {\Sigma }}}
2402:followed by a coordinate-by-coordinate
1245:are in descending order. In this case,
32385:
30850:
30274:
28212:has a set of orthonormal eigenvectors
27690:is the unique positive square root of
26871:
25542:{\displaystyle \mathbf {\Sigma } _{t}}
25277:{\displaystyle \mathbf {\Sigma } _{t}}
24845:{\displaystyle \mathbf {\Sigma } _{r}}
24640:{\displaystyle \mathbf {\Sigma } _{r}}
24209:{\displaystyle \mathbf {\Sigma } _{k}}
24058:{\displaystyle \mathbf {\Sigma } _{k}}
21013:the columns of the transformed matrix
20641:
19233:{\displaystyle S^{m-1}\times S^{n-1}.}
16010:Moreover, the second equation implies
15426:therefore contain the eigenvectors of
14918:the number of non-zero eigenvalues of
13302:(e.g., in genomic signal processing).
8262:corresponding to diagonal elements of
4143:are unitary, we know that the columns
3166:
910:, respectively. They form two sets of
196:with an orthonormal eigenbasis to any
32073:Spectral theory of normal C*-algebras
31871:Spectral theory of normal C*-algebras
31734:
31250:
31071:
29483:O. Alter; G. H. Golub (August 2006).
28717:{\displaystyle \{\mathbf {U} e_{i}\}}
28675:{\displaystyle \{\mathbf {U} e_{i}\}}
28069:as they have a discrete spectrum. If
28061:Singular values and compact operators
26766:{\displaystyle \mathbf {M} =(m_{ij})}
22723:{\displaystyle z_{i}\in \mathbb {C} }
21297:
20176:and the second equation from left by
19324:is also compact. Furthermore, since
18688:Based on variational characterization
18684:have the same non-zero eigenvalues).
17096:is a (generally not complete) set of
15015:is here by definition a matrix whose
13498:
13405:, in a form often referred to as the
13346:in natural-language text processing.
7273:
7262:
7255:
7248:
7234:
7218:
7207:
7200:
7193:
7182:
7169:
7162:
7155:
7148:
7137:
7128:
7121:
7114:
7107:
7100:
7091:
7084:
7077:
7070:
7063:
6458:
6451:
6444:
6437:
6423:
6414:
6407:
6400:
6393:
6386:
6377:
6370:
6363:
6353:
6346:
6330:
6323:
6316:
6309:
6295:
6286:
6279:
6269:
6262:
6255:
6207:
6198:
6170:
6137:
6100:
6043:
6033:
6026:
6019:
6007:
6000:
5993:
5986:
5977:
5970:
5963:
5953:
5944:
5937:
5927:
5920:
4703:{\displaystyle T(\mathbf {V} _{i})=0}
32068:Spectral theory of compact operators
30833:"§14.4 Singular Value Decomposition"
30222:
29648:
27180:are equal to the singular values of
27025:are equal to the singular values of
25345:{\displaystyle \mathbf {V} _{t}^{*}}
24913:{\displaystyle \mathbf {V} _{r}^{*}}
24277:{\displaystyle \mathbf {V} _{k}^{*}}
21134:{\displaystyle M\leftarrow J^{T}GMJ}
20969:is recovered as follows: the matrix
18083:and hence the overall dimensions of
16991:is a set of orthogonal vectors, and
13476:detection. A combination of SVD and
13285:
11076:denoting the conjugate transpose of
10623:Solving homogeneous linear equations
8893:Relation to eigenvalue decomposition
7746:are equal to the singular values of
7669:In any singular value decomposition
4086:
30370:
29005:or reflections. In 1970, Golub and
28905:in 1915, who arrived at it via the
27895:for operators: we can simply write
27796:with nontrivial kernel, a suitable
27213:Bounded operators on Hilbert spaces
26177:i.e. the largest singular value of
24111:{\displaystyle \mathbf {\Sigma } .}
21258:{\displaystyle U\leftarrow UG^{T}J}
15279:. This can be expressed by writing
14986:{\displaystyle \ell \leq \min(n,m)}
13821:when restricted to the unit sphere
12794:is the identity matrix, so that if
11959:
11233:{\displaystyle \|\mathbf {x} \|=1.}
5166:of this ellipsoid. Especially when
4655:{\displaystyle \mathbf {\Sigma } ,}
3934:are a basis of the column space of
2862:{\displaystyle \mathbf {\Sigma } ,}
124:, a scaling by the singular values
58:and the two canonical unit vectors
13:
32220:Cohen–Hewitt factorization theorem
29097:List of Fourier-related transforms
29072:Inequalities about singular values
28614:are called the singular values of
28009:is still a partial isometry while
22449:Thus, at every iteration, we have
21488:{\displaystyle \mathbf {\Sigma } }
21473:(found on the diagonal entries of
20403:
20062:
19958:
19948:
19922:
19912:
19881:
19871:
19862:
19814:with corresponding singular value
19043:
18709:
18098:{\displaystyle \mathbf {\Sigma } }
15089:, corresponding to the eigenvalue
14296:For every unit length eigenvector
14138:
14128:
14104:
14094:
14011:
13947:
13937:
13913:
13903:
13757:
13428:{\displaystyle \mathbf {\Sigma } }
13245:
13154:
13104:
12090:{\displaystyle A_{ij}=u_{i}v_{j}.}
11915:{\displaystyle \mathbf {\Sigma } }
11616:
11531:{\displaystyle \mathbf {\Sigma } }
10441:with singular value decomposition
10274:{\displaystyle \mathbf {\Sigma } }
9458:{\displaystyle \mathbf {\Sigma } }
9018:the following two relations hold:
8279:{\displaystyle \mathbf {\Sigma } }
7737:{\displaystyle \mathbf {\Sigma } }
6508:{\displaystyle \mathbf {\Sigma } }
6210:
6173:
6140:
6103:
4953:to a non-negative multiple of the
4812:one can find orthonormal bases of
3022:, with the singular values of any
2424:{\displaystyle \mathbf {\Sigma } }
2244:{\displaystyle \mathbf {\Sigma } }
1826:{\displaystyle \mathbf {\Sigma } }
1418:{\displaystyle \mathbf {\Sigma } }
1260:{\displaystyle \mathbf {\Sigma } }
1223:
739:{\displaystyle \mathbf {\Sigma } }
702:
660:
406:{\displaystyle \mathbf {\Sigma } }
14:
32429:
32225:Extensions of symmetric operators
31056:
30947:Strang G. (1998). "Section 6.7".
29768:Sarwar, Badrul; Karypis, George;
29635:10.1090/S0025-5718-1992-1106966-2
28067:compact operator on Hilbert space
27324:Namely, for any bounded operator
27125:for invertible diagonal matrices
22080:SVD problems, similar to how the
21816:The first step can be done using
20626:{\displaystyle \mathbf {v} _{1},}
19980:After some algebra, this becomes
19568:{\displaystyle \mathbf {v} _{1}.}
18847:where these maxima are attained.
18074:{\displaystyle \mathbf {U} _{2},}
16132:, into the following conditions:
14891:{\displaystyle \ell \times \ell }
13334:. It is also used in output-only
13305:
11922:except that it contains only the
9969:{\displaystyle \mathbf {U} _{i}.}
5373:{\displaystyle \mathbf {R} ^{n}.}
5153:{\displaystyle \mathbf {R} ^{m}.}
5089:{\displaystyle \mathbf {R} ^{n}.}
4056:are a basis of the null space of
3474:{\displaystyle \mathbf {U} ^{*},}
2970:{\displaystyle \mathbf {R} ^{m}.}
2833:{\displaystyle \mathbf {R} ^{n},}
2705:{\displaystyle \mathbf {R} ^{m}.}
2493:has a positive determinant, then
2151:{\displaystyle \mathbf {R} _{m},}
1496:{\displaystyle r\leq \min\{m,n\}}
1350:The term sometimes refers to the
1177:{\displaystyle r\leq \min\{m,n\}}
32043:Positive operator-valued measure
31715:
31714:
31641:Topological quantum field theory
30650:Chicco, D; Masseroli, M (2015).
30314:Chicco, D; Masseroli, M (2015).
30139:
30133:
30113:
30107:
30096:
30079:
29649:Shen, Jianhong (Jackie) (2001).
28989:, which uses plane rotations or
28985:in 1958, resembling closely the
28882:
28815:
28804:
28777:
28736:
28697:
28655:
28624:
28527:
28507:
28458:
28438:
28421:
28405:
28373:
28357:
28346:
28165:
28154:
28127:
28035:
28019:
27987:
27981:
27949:
27933:
27919:
27913:
27905:
27749:
27712:
27701:
27668:
27652:
27511:
27495:
27487:
27395:
27366:
27334:
27281:
27246:
27240:
27235:
27227:
27190:
27161:
27135:
27106:
27101:
27096:
27067:
27035:
27006:
26980:
26951:
26946:
26941:
26912:
26886:
26876:The singular values of a matrix
26737:
26708:
26665:
26560:
26521:
26510:
26483:
26475:
26459:
26422:
26411:
26388:
26380:
26309:
26298:
26255:
26244:
26223:
26187:
26140:
26129:
26083:
26072:
26022:
26011:
25966:
25955:
25940:
25911:
25792:
25757:
25700:
25630:
25620:the smallest singular values of
25601:
25529:
25485:{\displaystyle \mathbf {V} ^{*}}
25472:
25421:
25327:
25264:
25212:{\displaystyle \mathbf {U} _{t}}
25199:
25158:
25146:
25134:
25119:
25061:
25045:but rather provides the optimal
25026:
24962:In many applications the number
24957:
24895:
24832:
24780:{\displaystyle \mathbf {U} _{r}}
24767:
24699:{\displaystyle \mathbf {V} ^{*}}
24686:
24659:
24627:
24607:{\displaystyle \mathbf {V} ^{*}}
24594:
24543:
24479:
24467:
24455:
24446:
24420:
24383:
24259:
24196:
24144:{\displaystyle \mathbf {U} _{k}}
24131:
24101:
24045:
24015:
23989:
23946:{\displaystyle \mathbf {V} _{k}}
23933:
23913:{\displaystyle \mathbf {U} _{k}}
23900:
23815:
23803:
23791:
23782:
23756:
23698:
22740:
22610:
22592:
22521:
22513:
22478:
22472:
22467:
22459:
22424:
22418:
22410:
22384:
22354:
22349:
22341:
22284:
22271:
22261:
22254:
22217:
22206:
22171:
22165:
22139:
21955:to a triangular matrix with the
21939:
21872:{\displaystyle 4mn^{2}-4n^{3}/3}
21589:
21553:
21547:
21521:
21510:
21481:
21457:
21447:The non-zero singular values of
21428:
21417:
21390:
21355:
21349:
21316:
20610:
20590:{\displaystyle \mathbf {u} _{1}}
20577:
20550:
20524:
20495:
20487:
20439:
20410:
20398:
20379:
20350:
20344:
20241:
20227:
20187:
20147:
20107:
20069:
20057:
20032:
20000:
19994:
19964:
19953:
19928:
19917:
19892:
19887:
19876:
19798:
19779:{\displaystyle \mathbf {v} _{1}}
19766:
19746:{\displaystyle \mathbf {u} _{1}}
19733:
19708:{\displaystyle \mathbf {v} _{1}}
19695:
19675:{\displaystyle \mathbf {u} _{1}}
19662:
19632:
19624:
19552:
19532:{\displaystyle \mathbf {u} _{1}}
19519:
19421:
19358:
19121:
19078:
19054:
19049:
19038:
19026:
19018:
18997:{\displaystyle \mathbb {R} ^{k}}
18860:
18831:
18805:
18776:
18750:
18721:
18716:
18703:
18668:
18657:
18624:
18618:
18592:
18581:
18548:
18542:
18507:
18501:
18496:
18488:
18459:
18440:
18423:
18411:
18376:
18359:
18331:
18317:
18280:
18266:
18164:
18150:
18091:
18058:
18043:where extra zero rows are added
17973:
17948:
17938:to make it unitary. Now, define
17929:{\displaystyle \mathbf {V} _{2}}
17916:
17896:{\displaystyle \mathbf {V} _{1}}
17883:
17841:
17827:
17810:
17795:{\displaystyle \mathbf {U} _{2}}
17782:
17766:{\displaystyle \mathbf {U} _{1}}
17753:
17723:
17719:
17694:
17688:
17667:
17642:
17630:
17624:
17613:
17596:
17574:
17559:
17542:
17477:
17462:{\displaystyle \mathbf {U} _{1}}
17449:
17433:{\displaystyle \mathbf {V} _{1}}
17420:
17404:{\displaystyle \mathbf {U} _{1}}
17391:
17315:
17267:{\displaystyle \mathbf {U} _{1}}
17254:
17231:
17220:
17204:{\displaystyle \mathbf {V} _{2}}
17191:
17122:{\displaystyle \mathbf {V} _{1}}
17109:
17032:
16944:
16868:
16857:
16781:
16762:
16750:
16744:
16719:
16705:
16699:
16672:
16653:
16638:
16632:
16621:
16599:
16587:
16578:
16570:
16551:
16534:
16512:
16500:
16494:
16475:
16458:
16446:
16400:
16388:
16382:
16368:
16325:
16301:
16289:
16269:
16257:
16238:
16219:
16202:
16183:
16164:
16147:
16125:{\displaystyle \mathbf {V} _{2}}
16112:
16096:{\displaystyle \mathbf {V} _{1}}
16083:
16060:
16038:
16024:
16018:
15991:
15977:
15971:
15960:
15943:
15933:
15919:
15913:
15902:
15885:
15830:
15800:
15794:
15783:
15766:
15752:
15746:
15735:
15718:
15702:
15696:
15685:
15668:
15654:
15648:
15637:
15620:
15589:
15573:
15558:
15547:
15522:
15501:
15468:
15446:
15435:
15419:{\displaystyle \mathbf {V} _{2}}
15406:
15390:{\displaystyle \mathbf {V} _{1}}
15377:
15340:
15326:
15309:
15287:
15244:
15218:
15207:
15158:
15101:
15075:
15064:
15001:
14938:
14927:
14858:
14806:
14784:
14773:
14768:
14757:
14745:
14720:
14664:
14653:
14600:
14565:
14509:
14504:
14493:
14481:
14449:
14420:
14377:{\displaystyle f(\mathbf {v} ),}
14364:
14332:
14306:
14277:
14267:is a unit length eigenvector of
14251:
14222:
14211:
14206:
14177:
14169:
14158:
14144:
14133:
14115:
14110:
14099:
14069:
14041:
13953:
13942:
13924:
13919:
13908:
13878:
13835:
13805:
13768:
13763:
13752:
13739:
13638:
13608:
13579:
13568:
13563:
13537:
13421:
13251:
13240:
13231:
13168:
13160:
13149:
13126:
13118:
13113:
13092:
13063:
13037:
13011:
12976:
12967:
12958:
12953:
12945:
12915:
12909:
12901:
12869:
12863:
12855:
12823:
12817:
12812:
12804:
12778:
12746:
12740:
12735:
12700:
12694:
12665:
12657:
12624:
12598:
12569:
12459:
12433:
12374:{\displaystyle \mathbf {A} _{i}}
12361:
12286:{\displaystyle \mathbf {V} _{i}}
12273:
12253:{\displaystyle \mathbf {U} _{i}}
12240:
12204:
12189:
12154:
12135:
12109:
12020:
12012:
12004:
11974:
11908:
11880:
11844:
11832:
11824:
11810:
11778:
11732:
11686:
11657:
11595:
11568:
11524:
11500:
11465:
11446:{\displaystyle \mathbf {V} ^{*}}
11433:
11399:
11373:
11347:
11321:
11292:
11250:
11217:
11190:
11185:
11155:
11137:Total least squares minimization
11115:
11105:is called a left null vector of
11086:
11067:{\displaystyle \mathbf {x} ^{*}}
11054:
11027:
11019:
11008:
10981:
10954:
10924:
10898:
10872:
10843:
10813:
10787:
10761:
10735:
10706:
10680:
10654:
10646:
10641:
10597:
10569:
10535:
10523:
10517:
10503:
10470:
10464:
10459:
10451:
10425:
10406:
10345:
10319:
10293:
10267:
10215:
10189:
10134:
10099:
10093:
10085:
10053:
10047:
10042:
10034:
10005:
10000:
9992:
9953:
9933:{\displaystyle \mathbf {V} _{i}}
9920:
9822:
9816:
9811:
9803:
9742:
9685:
9659:
9627:
9621:
9616:
9608:
9563:
9525:
9519:
9493:
9482:
9451:
9414:
9408:
9382:
9351:
9340:
9309:
9266:
9251:
9245:
9237:
9223:
9211:
9205:
9193:
9187:
9182:
9164:
9158:
9140:
9131:
9120:
9111:
9097:
9091:
9086:
9074:
9062:
9056:
9044:
9033:
8993:
8987:
8982:
8974:
8948:
8871:
8845:
8819:
8793:
8767:
8741:
8715:
8689:
8626:
8600:
8520:
8431:
8358:cannot be the same dimension if
8335:
8272:
8246:
8220:
8200:{\displaystyle \mathbf {u} _{2}}
8187:
8167:{\displaystyle \mathbf {u} _{1}}
8154:
8117:
8060:
8029:
7972:
7959:It is always possible to find a
7916:
7856:
7830:
7756:
7730:
7698:
7692:
7687:
7679:
7615:
7589:
7556:
7541:
7530:
7517:
7502:
7499:
7438:
7381:
7355:
7042:
7015:
6979:
6810:
6804:
6789:
6667:
6661:
6634:{\displaystyle \mathbf {V} ^{*}}
6621:
6594:
6567:{\displaystyle \mathbf {V} ^{*}}
6554:
6527:
6501:
6230:
6200:
6061:
5901:
5866:
5860:
5855:
5706:
5610:
5601:
5593:
5531:
5463:
5454:
5395:
5357:
5337:{\displaystyle \mathbf {V} ^{*}}
5324:
5280:{\displaystyle \mathbf {R} ^{n}}
5267:
5137:
5073:
4803:{\displaystyle T:K^{n}\to K^{m}}
4681:
4645:
4515:
4487:
4444:
4322:
4290:
4269:
4207:
4175:
4154:
4127:
4101:
4066:
4040:
3981:
3958:{\displaystyle \mathbf {M} ^{*}}
3945:
3918:
3872:{\displaystyle \mathbf {M} ^{*}}
3859:
3828:
3769:
3739:
3672:
3643:
3617:
3591:
3558:
3535:{\displaystyle \mathbf {V} ^{*}}
3522:
3492:
3458:
3429:
3400:
3381:{\displaystyle \mathbf {U} ^{*}}
3368:
3332:
3302:{\displaystyle \mathbf {V} _{i}}
3289:
3262:
3235:{\displaystyle \mathbf {V} ^{*}}
3222:
3195:
2954:
2934:{\displaystyle \mathbf {R} ^{n}}
2921:
2852:
2817:
2797:{\displaystyle \mathbf {R} ^{m}}
2784:
2764:{\displaystyle \mathbf {V} ^{*}}
2751:
2724:
2689:
2669:{\displaystyle \mathbf {R} ^{n}}
2656:
2566:
2543:{\displaystyle \mathbf {V} ^{*}}
2530:
2503:
2477:
2447:
2417:
2391:{\displaystyle \mathbf {V} ^{*}}
2378:
2348:{\displaystyle \mathbf {R} ^{m}}
2335:
2281:{\displaystyle \mathbf {x} _{i}}
2268:
2237:
2210:{\displaystyle \mathbf {V} ^{*}}
2197:
2170:
2135:
2108:
2105:
2097:
2065:
2012:{\displaystyle \mathbf {V} ^{*}}
1999:
1972:
1912:
1819:
1789:, which distorts the disk to an
1697:
1688:
1677:
1668:
1657:
1601:
1544:
1515:
1411:
1379:
1376:
1373:
1364:
1329:
1303:
1277:
1253:
1194:
1113:
1101:
1061:
999:
978:
945:
924:
894:
868:
842:
816:
786:
756:
732:
654:
648:
643:
615:
589:
563:
537:
516:{\displaystyle \mathbf {V} ^{*}}
503:
452:
399:
341:
312:
308:
304:
301:
293:
269:
32327:Rayleigh–Faber–Krahn inequality
30816:(3rd ed.). Johns Hopkins.
30552:
30539:
30493:
30453:
30423:
30364:
30307:
30268:
30257:
30246:
30235:
30216:
30055:
29922:
29869:
29812:
29761:
29708:
29675:
29642:
29598:
28277:{\displaystyle \{\sigma _{i}\}}
25661:
24747:{\displaystyle r\ll \min(m,n).}
24360:{\displaystyle k\ll \max(m,n).}
23608:
20813:become orthogonal. The indices
17245:with vanishing eigenvalue, and
16052:. Finally, the unitary-ness of
15940:
13146:
13140:
10205:is not necessarily unitary and
4751:{\displaystyle i>\min(m,n).}
4528:
2582:is real but not square, namely
285:is a factorization of the form
30949:Introduction to Linear Algebra
30909:. Cambridge University Press.
30884:. Cambridge University Press.
30191:
30123:
30074:
29688:Applied Mathematical Modelling
29535:
29476:
29417:
29358:
29277:
29266:
29231:
29188:
29170:von Neumann's trace inequality
29056:Empirical orthogonal functions
29051:Eigendecomposition of a matrix
29001:published in 1965, which uses
27631:{\displaystyle L^{2}(X,\mu ).}
27622:
27610:
27453:and a non-negative measurable
27435:
27423:
26867:Variations and generalizations
26832:
26813:
26760:
26744:
26260:
26239:
26145:
26124:
26027:
26006:
25123:
25083:by any matrix of a fixed rank
25065:
24738:
24726:
24405:
24351:
24339:
23873:
23861:
23579:
23544:
23532:
23497:
23485:
23450:
23438:
23403:
23391:
23356:
23344:
23309:
23291:
23275:
23261:
23245:
23231:
23215:
23201:
23185:
23158:
23083:
23055:
23039:
23025:
23009:
22995:
22979:
22965:
22949:
22926:
22910:
22896:
22880:
22866:
22850:
22836:
22820:
22517:
22463:
22414:
22345:
22003:{\displaystyle 2mn^{2}+2n^{3}}
21794:
21778:
21751:
21745:
21718:
21712:
21640:
21624:
21380:The right-singular vectors of
21287:{\displaystyle V\leftarrow VJ}
21275:
21236:
21109:
20894:
20852:
20832:
20820:
20766:{\displaystyle J(p,q,\theta )}
20760:
20742:
20732:of the Jacobi rotation matrix
20694:
20676:
20667:
20499:
20483:
19636:
19620:
19477:This largest value is denoted
19030:
19014:
18962:
18950:
18906:matrix with real entries. Let
17344:
17061:
16848:is the set of eigenvectors of
16648:
16628:
16614:
16574:
15248:
15105:
14980:
14968:
14952:(which can be shown to verify
14788:
14485:
14477:
14368:
14360:
14173:
14154:
13747:
13726:
13213:{\displaystyle \|\cdot \|_{F}}
11884:
11836:
11814:
11736:
11690:
11599:
9261:
9241:
9135:
9115:
8931:can only be applied to square
7805:
7793:
7306:SVD and spectral decomposition
5565:
5559:
5503:
5497:
5429:
5423:
5241:
5235:
4787:
4742:
4730:
4691:
4676:
4562:
4550:
4497:
4482:
4440:
4416:
4369:(with respect to the standard
4338:yield an orthonormal basis of
2101:
1779:in blue together with the two
1:
32235:Limiting absorption principle
31437:Uniform boundedness principle
30606:
29668:10.1016/S0024-3795(00)00322-0
28892:{\displaystyle \mathbf {A} .}
28746:{\displaystyle \mathbf {M} .}
28634:{\displaystyle \mathbf {M} .}
27405:{\displaystyle \mathbf {V} ,}
27376:{\displaystyle \mathbf {U} ,}
27344:{\displaystyle \mathbf {M} ,}
27297:on a separable Hilbert space
27200:{\displaystyle \mathbf {A} .}
27171:{\displaystyle \mathbf {E} ,}
27077:{\displaystyle \mathbf {A} .}
27045:{\displaystyle \mathbf {A} .}
27016:{\displaystyle \mathbf {V} ,}
26922:{\displaystyle \mathbf {A} .}
26718:{\displaystyle \mathbf {M} .}
26675:{\displaystyle \mathbf {M} .}
26197:{\displaystyle \mathbf {M} .}
26115:is the largest eigenvalue of
25767:{\displaystyle \mathbf {M} .}
25649:Truncated SVD is employed in
25611:{\displaystyle \mathbf {M} ,}
25047:low-rank matrix approximation
25036:{\displaystyle \mathbf {M} ,}
24393:{\displaystyle \mathbf {M} ,}
24025:{\displaystyle \mathbf {V} ,}
23623:not corresponding to rows of
21306:The left-singular vectors of
18786:{\displaystyle \mathbf {V} ,}
18478:which is the desired result:
17129:the matrix whose columns are
14588:Based on the spectral theorem
14525:is a real-valued function of
14430:{\displaystyle \mathbf {M} .}
14410:is the largest eigenvalue of
14287:{\displaystyle \mathbf {M} .}
13073:{\displaystyle \mathbf {B} .}
12999:orthogonal Procrustes problem
12634:{\displaystyle \mathbf {A} .}
11788:{\displaystyle \mathbf {M} ,}
11550:Low-rank matrix approximation
11475:{\displaystyle \mathbf {U} .}
11409:{\displaystyle \mathbf {M} .}
11302:{\displaystyle \mathbf {M} .}
11125:{\displaystyle \mathbf {A} .}
11096:{\displaystyle \mathbf {x} ,}
10853:{\displaystyle \mathbf {A} .}
10716:{\displaystyle \mathbf {x} .}
10355:{\displaystyle \mathbf {M} .}
10144:{\displaystyle \mathbf {M} ,}
9869:{\displaystyle e^{i\varphi }}
8881:{\displaystyle \mathbf {M} .}
8670:{\displaystyle e^{i\varphi }}
8590:exists, the extra columns of
8345:{\displaystyle \mathbf {M} ,}
8127:{\displaystyle \mathbf {M} .}
8039:{\displaystyle \mathbf {M} .}
7766:{\displaystyle \mathbf {M} .}
3653:{\displaystyle \mathbf {M} .}
3502:{\displaystyle \mathbf {V} ,}
3439:{\displaystyle \mathbf {U} ,}
3410:{\displaystyle \mathbf {V} ,}
3311:to the stretched unit vector
2075:{\displaystyle \mathbf {A} ,}
1783:. We then see the actions of
1525:{\displaystyle \mathbf {M} ,}
1339:{\displaystyle \mathbf {M} .}
1204:{\displaystyle \mathbf {M} .}
631:can be guaranteed to be real
226:matrix. It is related to the
32393:Singular value decomposition
31861:Singular value decomposition
29569:10.1371/journal.pone.0078913
29321:. IEEE. pp. 2691–2695.
29209:10.1016/0166-2236(95)94496-R
29138:Principal component analysis
28784:{\displaystyle \mathbf {M} }
28134:{\displaystyle \mathbf {M} }
27756:{\displaystyle \mathbf {U} }
27288:{\displaystyle \mathbf {M} }
27142:{\displaystyle \mathbf {D} }
26987:{\displaystyle \mathbf {U} }
26893:{\displaystyle \mathbf {A} }
26788:
26574:
25918:{\displaystyle \mathbf {M} }
25799:{\displaystyle \mathbf {M} }
25707:{\displaystyle \mathbf {M} }
25637:{\displaystyle \mathbf {M} }
25428:{\displaystyle \mathbf {U} }
24666:{\displaystyle \mathbf {U} }
24550:{\displaystyle \mathbf {U} }
24427:{\displaystyle \mathbf {M} }
24410:The compact SVD of a matrix
23996:{\displaystyle \mathbf {U} }
23882:{\displaystyle k=\min(m,n),}
23763:{\displaystyle \mathbf {M} }
23705:{\displaystyle \mathbf {M} }
22756:is the identity matrix, and
22747:{\displaystyle \mathbf {I} }
22550:Analytic result of 2 × 2 SVD
22391:{\displaystyle \mathbf {R} }
22146:{\displaystyle \mathbf {M} }
22124:Trefethen & Bau III 1997
22014:Trefethen & Bau III 1997
21946:{\displaystyle \mathbf {M} }
21811:Trefethen & Bau III 1997
21596:{\displaystyle \mathbf {M} }
21464:{\displaystyle \mathbf {M} }
21397:{\displaystyle \mathbf {M} }
21323:{\displaystyle \mathbf {M} }
21226:are accumulated as follows:
20557:{\displaystyle \mathbf {v} }
20531:{\displaystyle \mathbf {u} }
19839:{\displaystyle \sigma _{1}.}
19805:{\displaystyle \mathbf {M} }
19428:{\displaystyle \mathbf {v} }
19365:{\displaystyle \mathbf {u} }
18867:{\displaystyle \mathbf {M} }
18838:{\displaystyle \mathbf {V} }
18812:{\displaystyle \mathbf {U} }
18757:{\displaystyle \mathbf {U} }
18740:considered as a function of
17484:{\displaystyle \mathbf {D} }
17341:
17058:
16067:{\displaystyle \mathbf {V} }
15475:{\displaystyle \mathbf {V} }
15294:{\displaystyle \mathbf {V} }
15165:{\displaystyle \mathbf {V} }
15008:{\displaystyle \mathbf {V} }
14865:{\displaystyle \mathbf {D} }
14727:{\displaystyle \mathbf {V} }
14607:{\displaystyle \mathbf {M} }
14572:{\displaystyle \mathbf {M} }
14456:{\displaystyle \mathbf {u} }
14339:{\displaystyle \mathbf {M} }
14313:{\displaystyle \mathbf {v} }
14258:{\displaystyle \mathbf {u} }
14076:{\displaystyle \mathbf {M} }
14048:{\displaystyle \mathbf {x} }
13885:{\displaystyle \mathbf {u} }
13812:{\displaystyle \mathbf {u} }
13645:{\displaystyle \mathbf {M} }
13615:{\displaystyle \mathbf {M} }
13544:{\displaystyle \mathbf {M} }
13440:numerical weather prediction
13320:principal component analysis
13220:denotes the Frobenius norm.
13044:{\displaystyle \mathbf {A} }
13018:{\displaystyle \mathbf {O} }
12785:{\displaystyle \mathbf {I} }
12684:The solution is the product
12605:{\displaystyle \mathbf {O} }
12576:{\displaystyle \mathbf {A} }
12466:{\displaystyle \mathbf {V} }
12440:{\displaystyle \mathbf {U} }
12116:{\displaystyle \mathbf {M} }
11981:{\displaystyle \mathbf {A} }
11664:{\displaystyle \mathbf {M} }
11619:, which has a specific rank
11575:{\displaystyle \mathbf {M} }
11507:{\displaystyle \mathbf {M} }
11419:
11380:{\displaystyle \mathbf {M} }
11354:{\displaystyle \mathbf {M} }
11328:{\displaystyle \mathbf {M} }
11257:{\displaystyle \mathbf {A} }
11162:{\displaystyle \mathbf {x} }
10988:{\displaystyle \mathbf {x} }
10961:{\displaystyle \mathbf {x} }
10931:{\displaystyle \mathbf {A} }
10905:{\displaystyle \mathbf {A} }
10879:{\displaystyle \mathbf {x} }
10820:{\displaystyle \mathbf {A} }
10794:{\displaystyle \mathbf {x} }
10768:{\displaystyle \mathbf {x} }
10742:{\displaystyle \mathbf {A} }
10725:A typical situation is that
10687:{\displaystyle \mathbf {A} }
10629:homogeneous linear equations
10432:{\displaystyle \mathbf {M} }
10326:{\displaystyle \mathbf {V} }
10300:{\displaystyle \mathbf {U} }
10222:{\displaystyle \mathbf {D} }
10196:{\displaystyle \mathbf {U} }
9909:to either its corresponding
9749:{\displaystyle \mathbf {M} }
9692:{\displaystyle \mathbf {D} }
9666:{\displaystyle \mathbf {U} }
9583:, and thus also square, the
9570:{\displaystyle \mathbf {M} }
9389:{\displaystyle \mathbf {U} }
9316:{\displaystyle \mathbf {V} }
8955:{\displaystyle \mathbf {M} }
8852:{\displaystyle \mathbf {V} }
8826:{\displaystyle \mathbf {U} }
8800:{\displaystyle \mathbf {V} }
8774:{\displaystyle \mathbf {U} }
8748:{\displaystyle \mathbf {V} }
8722:{\displaystyle \mathbf {U} }
8696:{\displaystyle \mathbf {M} }
8633:{\displaystyle \mathbf {V} }
8607:{\displaystyle \mathbf {U} }
8527:{\displaystyle \mathbf {V} }
8438:{\displaystyle \mathbf {U} }
8253:{\displaystyle \mathbf {V} }
8227:{\displaystyle \mathbf {U} }
8067:{\displaystyle \mathbf {V} }
7979:{\displaystyle \mathbf {U} }
7923:{\displaystyle \mathbf {M} }
7863:{\displaystyle \mathbf {V} }
7837:{\displaystyle \mathbf {U} }
7622:{\displaystyle \mathbf {v} }
7596:{\displaystyle \mathbf {u} }
7445:{\displaystyle \mathbf {v} }
7388:{\displaystyle \mathbf {u} }
7362:{\displaystyle \mathbf {M} }
7022:{\displaystyle \mathbf {V} }
6643:are real valued, each is an
6601:{\displaystyle \mathbf {U} }
6534:{\displaystyle \mathbf {U} }
5547:to this ellipsoid to obtain
5538:{\displaystyle \mathbf {U} }
5402:{\displaystyle \mathbf {D} }
4329:{\displaystyle \mathbf {V} }
4214:{\displaystyle \mathbf {U} }
4134:{\displaystyle \mathbf {V} }
4108:{\displaystyle \mathbf {U} }
4073:{\displaystyle \mathbf {M} }
4047:{\displaystyle \mathbf {V} }
3988:{\displaystyle \mathbf {M} }
3925:{\displaystyle \mathbf {V} }
3835:{\displaystyle \mathbf {U} }
3776:{\displaystyle \mathbf {M} }
3746:{\displaystyle \mathbf {U} }
3679:{\displaystyle \mathbf {M} }
3624:{\displaystyle \mathbf {V} }
3598:{\displaystyle \mathbf {U} }
3565:{\displaystyle \mathbf {M} }
3269:{\displaystyle \mathbf {M} }
3202:{\displaystyle \mathbf {U} }
2986:As shown in the figure, the
2731:{\displaystyle \mathbf {U} }
2573:{\displaystyle \mathbf {M} }
2510:{\displaystyle \mathbf {U} }
2484:{\displaystyle \mathbf {M} }
2454:{\displaystyle \mathbf {U} }
2315:{\displaystyle \sigma _{i}.}
2177:{\displaystyle \mathbf {U} }
1979:{\displaystyle \mathbf {U} }
1919:{\displaystyle \mathbf {M} }
1608:{\displaystyle \mathbf {V} }
1551:{\displaystyle \mathbf {U} }
1319:) is uniquely determined by
1310:{\displaystyle \mathbf {V} }
1284:{\displaystyle \mathbf {U} }
1238:{\displaystyle \Sigma _{ii}}
901:{\displaystyle \mathbf {M} }
875:{\displaystyle \mathbf {V} }
849:{\displaystyle \mathbf {U} }
823:{\displaystyle \mathbf {M} }
793:{\displaystyle \mathbf {M} }
763:{\displaystyle \mathbf {M} }
622:{\displaystyle \mathbf {V} }
596:{\displaystyle \mathbf {U} }
570:{\displaystyle \mathbf {M} }
544:{\displaystyle \mathbf {V} }
494:complex unitary matrix, and
459:{\displaystyle \mathbf {V} }
348:{\displaystyle \mathbf {U} }
276:{\displaystyle \mathbf {M} }
167:singular value decomposition
7:
32292:Hearing the shape of a drum
31975:Decomposition of a spectrum
29327:10.1109/ICASSP.2018.8462274
29012:
29003:Householder transformations
28987:Jacobi eigenvalue algorithm
28960:{\displaystyle \sigma _{k}}
28605:{\displaystyle \sigma _{i}}
28093:is compact, every non-zero
27827:may not be found such that
26655:are the singular values of
26646:{\displaystyle \sigma _{i}}
25690:largest singular values of
23741:
22776:{\displaystyle \sigma _{i}}
22082:Jacobi eigenvalue algorithm
21063:Jacobi eigenvalue algorithm
20468:This proves the statement.
19599:{\displaystyle \sigma _{1}}
19499:{\displaystyle \sigma _{1}}
13435:matrix is larger than one.
12937:of the Polar Decomposition
12405:{\displaystyle \sigma _{i}}
12341:{\displaystyle \sigma _{i}}
9900:{\displaystyle \sigma _{i}}
9784:{\displaystyle \sigma _{i}}
9723:{\displaystyle \sigma _{i}}
7811:{\displaystyle p=\min(m,n)}
7315:A non-negative real number
5382:On a second move, apply an
4602:{\displaystyle \sigma _{i}}
3248:, which can be regarded as
2901:{\displaystyle \min\{m,n\}}
2365:: a rotation or reflection
1427:is square diagonal of size
1044:{\displaystyle \sigma _{i}}
746:are uniquely determined by
440:rectangular diagonal matrix
10:
32434:
31880:Special Elements/Operators
31580:Invariant subspace problem
31134:System of linear equations
30020:Advances in Space Research
29747:10.1103/PhysRevD.99.024010
29102:Locality-sensitive hashing
28835:
28793:is compact if and only if
25378:{\displaystyle t\times n.}
25245:{\displaystyle m\times t,}
24946:{\displaystyle r\times n.}
24813:{\displaystyle m\times r,}
24310:{\displaystyle k\times n.}
24177:{\displaystyle m\times k,}
22323:to find the real diagonal
21057:Two-sided Jacobi algorithm
20927:is the number of columns.
20647:One-sided Jacobi algorithm
18608:(This shows directly that
15191:{\displaystyle j>\ell }
11708:under the constraint that
11647:of the difference between
11418:For example, in the above
11270:Range, null space and rank
9732:along the diagonal. When
5661:
2871:besides scaling the first
1451:{\displaystyle r\times r,}
1354:, a similar decomposition
32352:Superstrong approximation
32274:
32258:
32215:Banach algebra cohomology
32202:
32166:
32135:
32081:
32048:Projection-valued measure
32033:Borel functional calculus
32025:
31967:
31924:
31879:
31833:
31805:Projection-valued measure
31772:
31710:
31669:
31593:
31572:
31531:
31470:
31412:
31358:
31300:
31293:
31203:
31185:Cache-oblivious algorithm
31167:
31126:
31105:
30907:Topics in Matrix Analysis
30704:; Bau III, David (1997).
30668:10.1109/TCBB.2014.2382127
30332:10.1109/TCBB.2014.2382127
30225:SIAM J. Sci. Stat. Comput
30041:10.1016/j.asr.2021.10.028
29996:10.1007/s42064-022-0147-z
29931:Intelligent Data Analysis
29701:10.1016/j.apm.2013.04.025
29302:10.1016/j.dsp.2015.10.011
29290:Digital Signal Processing
29252:10.1152/jn.2001.85.3.1220
29041:Digital signal processing
28240:{\displaystyle \{e_{i}\}}
27733:Borel functional calculus
27444:{\displaystyle (X,\mu ),}
26358:{\displaystyle n\times n}
25997:But, in the matrix case,
25893:{\displaystyle \ell ^{2}}
25307:{\displaystyle t\times t}
24875:{\displaystyle r\times r}
24239:{\displaystyle k\times k}
23679:{\displaystyle m\times n}
22575:{\displaystyle 2\times 2}
22554:The singular values of a
22116:Golub & Van Loan 1996
22105:{\displaystyle 2\times 2}
22071:{\displaystyle 2\times 2}
21800:{\displaystyle O(mn^{2})}
21733:iterations, each costing
21646:{\displaystyle O(mn^{2})}
20962:{\displaystyle M=USV^{T}}
18897:{\displaystyle m\times n}
18124:{\displaystyle m\times n}
14703:{\displaystyle n\times n}
14635:{\displaystyle m\times n}
13993:{\displaystyle \lambda .}
13675:{\displaystyle n\times n}
12555:Nearest orthogonal matrix
12099:Specifically, the matrix
11145:problem seeks the vector
10389:{\displaystyle m\times n}
9598:, and thus decomposed as
9441:The non-zero elements of
8918:{\displaystyle m\times n}
7956:distinct singular values.
7897:{\displaystyle m\times n}
5687:{\displaystyle 4\times 5}
5122:maps this sphere onto an
3104:{\displaystyle m\times n}
3043:{\displaystyle n\times n}
2603:{\displaystyle m\times n}
2042:{\displaystyle m\times m}
2021:can be chosen to be real
1949:{\displaystyle m\times m}
1902:In the special case when
1758:Intuitive interpretations
1636:{\displaystyle n\times r}
1581:{\displaystyle m\times r}
487:{\displaystyle n\times n}
432:{\displaystyle m\times n}
378:{\displaystyle m\times m}
252:{\displaystyle m\times n}
217:{\displaystyle m\times n}
32403:Numerical linear algebra
31944:Spectrum of a C*-algebra
31815:Spectrum of a C*-algebra
31549:Spectrum of a C*-algebra
31236:General purpose software
31099:Numerical linear algebra
30706:Numerical linear algebra
29392:10.1073/pnas.97.18.10101
29181:
29087:Latent semantic indexing
29082:Latent semantic analysis
29046:Dimensionality reduction
28108:{\displaystyle \lambda }
25656:
25651:latent semantic indexing
25518:largest singular values
24067:contains only the first
22044:Jacobi orthogonalization
22025:Golub & Kahan (1965)
21679:{\displaystyle m\geq n.}
20566:which are orthogonal to
19608:is the largest value of
19468:{\displaystyle S^{n-1}.}
16074:translates, in terms of
15482:, the equation becomes:
14401:{\displaystyle \lambda }
13518:{\displaystyle \lambda }
13344:latent semantic indexing
13027:which most closely maps
10751:is known and a non-zero
10589:is the pseudoinverse of
9649:for some unitary matrix
8929:eigenvalue decomposition
8382:{\displaystyle m\neq n.}
8306:{\displaystyle \sigma .}
8288:all with the same value
7720:the diagonal entries of
7657:{\displaystyle \sigma ,}
2636:{\displaystyle m\neq n,}
32372:Wiener–Khinchin theorem
32307:Kuznetsov trace formula
32282:Almost Mathieu operator
32100:Banach function algebra
32089:Amenable Banach algebra
31846:Gelfand–Naimark theorem
31800:Noncommutative topology
31646:Noncommutative geometry
30966:Stewart, G. W. (1993).
30631:Bisgard, James (2021).
30291:10.1137/1.9780898719581
29785:University of Minnesota
29510:10.1073/pnas.0604756103
29451:10.1073/pnas.0406767101
29123:Nearest neighbor search
29036:Curse of dimensionality
29030:Correspondence analysis
27888:is a unitary operator.
26446:So the induced norm is
25575:{\displaystyle t\ll r,}
23955:contain only the first
22033:Demmel & Kahan 1990
21818:Householder reflections
20725:{\displaystyle \theta }
19402:{\displaystyle S^{m-1}}
19339:{\displaystyle \sigma }
19307:{\displaystyle S^{n-1}}
19270:{\displaystyle S^{m-1}}
19174:{\displaystyle \sigma }
18934:{\displaystyle S^{k-1}}
15368:, where the columns of
15198:, is an eigenvector of
14581:is no longer required.
14017:{\displaystyle \nabla }
13493:orbital station-keeping
13338:, where the non-scaled
13324:correspondence analysis
11337:span the null space of
10402:Applications of the SVD
9587:ensures that it can be
9553:In the special case of
8933:diagonalizable matrices
8501:{\displaystyle m<n,}
7330:{\displaystyle \sigma }
1861:of the ellipse are the
32347:Sturm–Liouville theory
32245:Sherman–Takeda theorem
32125:Tomita–Takesaki theory
31900:Hermitian/Self-adjoint
31851:Gelfand representation
31702:Tomita–Takesaki theory
31677:Approximation property
31621:Calculus of variations
30547:Golub & Kahan 1965
30398:10.1073/pnas.37.11.760
30275:Demmel, James (2000).
30207:
30163:
29857:Cite journal requires
29800:Cite journal requires
29605:Edelman, Alan (1992).
29107:Low-rank approximation
28961:
28893:
28866:James Joseph Sylvester
28842:differential geometers
28823:
28785:
28747:
28718:
28676:
28635:
28606:
28575:
28548:
28329:
28302:
28278:
28241:
28204:
28173:
28135:
28109:
28085:
28049:
28001:
27963:
27880:
27819:
27788:
27757:
27737:self-adjoint operators
27723:
27682:
27632:
27582:
27556:
27525:
27469:
27445:
27406:
27377:
27345:
27316:
27289:
27260:
27201:
27172:
27143:
27117:
27078:
27046:
27017:
26988:
26962:
26923:
26894:
26851:
26767:
26719:
26676:
26647:
26616:
26539:
26438:
26359:
26317:
26281:
26198:
26169:
26109:
26048:
25989:
25919:
25894:
25865:
25864:{\displaystyle K^{n}.}
25831:
25800:
25768:
25739:
25708:
25682:
25638:
25612:
25576:
25543:
25510:
25486:
25453:
25429:
25403:
25379:
25346:
25308:
25278:
25246:
25213:
25180:
25099:
25075:
25037:
25008:
25007:{\displaystyle t\ll r}
24978:
24947:
24914:
24876:
24846:
24814:
24781:
24748:
24700:
24667:
24641:
24608:
24575:
24551:
24525:
24501:
24428:
24394:
24361:
24311:
24278:
24240:
24210:
24178:
24145:
24112:
24083:
24059:
24026:
23997:
23971:
23947:
23914:
23883:
23837:
23764:
23730:
23706:
23680:
23654:
23600:
22777:
22748:
22724:
22687:
22576:
22529:
22495:
22441:
22392:
22362:
22303:
22228:
22188:
22147:
22106:
22072:
22040:GNU Scientific Library
22004:
21947:
21921:
21897:
21873:
21801:
21758:
21725:
21680:
21647:
21597:
21567:
21529:
21489:
21465:
21436:
21398:
21369:
21324:
21288:
21259:
21220:
21200:
21177:
21157:
21135:
21083:
21047:
21027:
21003:
20983:
20963:
20921:
20901:
20845:are swept cyclically,
20839:
20807:
20787:
20767:
20726:
20704:
20627:
20591:
20558:
20532:
20506:
20460:
20322:
20258:
20208:
20168:
20128:
19972:
19840:
19806:
19780:
19747:
19709:
19676:
19643:
19600:
19569:
19533:
19500:
19469:
19429:
19403:
19366:
19340:
19308:
19271:
19234:
19175:
19159:Consider the function
19151:
19108:
19065:
18998:
18969:
18935:
18898:
18868:
18839:
18813:
18787:
18758:
18732:
18676:
18638:
18600:
18562:
18524:
18470:
18125:
18099:
18075:
18035:
17930:
17897:
17863:
17796:
17767:
17736:
17525:
17505:
17485:
17463:
17434:
17405:
17373:
17268:
17239:
17205:
17176:
17123:
17090:
16985:
16927:
16876:
16842:
16789:
16730:
16683:
16427:
16346:
16126:
16097:
16068:
16046:
16002:
15866:
15476:
15454:
15420:
15391:
15362:
15295:
15273:
15226:
15192:
15166:
15144:
15124:
15083:
15049:
15029:
15009:
14987:
14946:
14912:
14892:
14866:
14842:
14728:
14704:
14672:
14644:complex matrix. Since
14636:
14608:
14573:
14544:
14517:
14457:
14431:
14402:
14378:
14340:
14314:
14288:
14259:
14233:
14188:
14077:
14059:Using the symmetry of
14049:
14018:
13994:
13967:
13894:necessarily satisfies
13886:
13858:
13813:
13783:
13676:
13646:
13616:
13590:
13545:
13519:
13456:radial basis functions
13429:
13392:
13259:
13214:
13179:
13074:
13045:
13019:
12984:
12929:
12883:
12837:
12786:
12760:
12717:
12676:
12635:
12606:
12577:
12543:
12467:
12441:
12406:
12375:
12342:
12311:
12287:
12254:
12221:
12125:can be decomposed as,
12117:
12091:
12031:
11982:
11938:
11916:
11900:is the same matrix as
11894:
11861:
11789:
11762:
11700:
11665:
11635:
11609:
11576:
11532:
11508:
11486:As a consequence, the
11476:
11447:
11410:
11381:
11355:
11329:
11303:
11258:
11234:
11198:
11163:
11126:
11097:
11068:
11035:
10989:
10962:
10932:
10906:
10880:
10854:
10821:
10795:
10769:
10743:
10717:
10688:
10662:
10605:
10583:
10552:
10484:
10433:
10390:
10356:
10327:
10301:
10275:
10249:
10223:
10197:
10171:
10145:
10113:
10067:
10016:
9970:
9934:
9901:
9870:
9836:
9785:
9760:positive semi-definite
9750:
9724:
9701:with complex elements
9693:
9667:
9641:
9571:
9542:
9501:
9459:
9431:
9390:
9362:
9317:
9287:
9010:
8956:
8919:
8882:
8853:
8827:
8801:
8775:
8749:
8723:
8697:
8671:
8634:
8608:
8582:
8558:
8528:
8502:
8469:
8439:
8413:
8412:{\displaystyle m>n}
8383:
8346:
8307:
8280:
8254:
8228:
8201:
8168:
8128:
8099:
8068:
8040:
8011:
7980:
7948:
7924:
7898:
7864:
7838:
7812:
7767:
7738:
7712:
7658:
7637:right-singular vectors
7623:
7597:
7571:
7477:
7446:
7420:
7389:
7363:
7331:
7294:
7023:
6997:
6635:
6602:
6568:
6535:
6509:
6483:
5880:
5837:
5688:
5651:
5624:
5575:
5539:
5513:
5477:
5436:
5403:
5374:
5338:
5305:
5281:
5248:
5215:
5191:
5154:
5114:
5090:
5054:
5027:
5003:
5002:{\displaystyle K^{m},}
4969:
4945:
4914:
4890:
4866:
4835:
4804:
4752:
4704:
4656:
4635:-th diagonal entry of
4627:
4603:
4572:
4462:
4361:
4330:
4304:
4250:
4215:
4189:
4135:
4109:
4074:
4048:
4022:
3989:
3959:
3926:
3900:
3873:
3836:
3810:
3777:
3747:
3721:
3680:
3654:
3625:
3599:
3566:
3536:
3503:
3475:
3440:
3411:
3382:
3349:
3303:
3278:maps the basis vector
3270:
3236:
3203:
3157:
3129:
3105:
3071:
3044:
3010:
2971:
2935:
2902:
2863:
2834:
2798:
2765:
2732:
2706:
2670:
2637:
2604:
2574:
2544:
2511:
2485:
2455:
2425:
2392:
2349:
2316:
2282:
2245:
2211:
2178:
2152:
2116:
2076:
2043:
2013:
1980:
1950:
1920:
1894:
1886:
1827:
1781:canonical unit vectors
1714:
1637:
1609:
1582:
1552:
1526:
1497:
1452:
1419:
1393:
1340:
1311:
1285:
1261:
1239:
1205:
1178:
1135:
1088:
1045:
1016:
959:
902:
876:
850:
824:
794:
764:
740:
718:
673:
623:
597:
571:
545:
517:
488:
460:
433:
407:
379:
349:
325:
277:
253:
218:
158:
32413:Matrix decompositions
31841:Gelfand–Mazur theorem
31697:Banach–Mazur distance
31660:Generalized functions
31231:Specialized libraries
31144:Matrix multiplication
31139:Matrix decompositions
31063:Online SVD calculator
30568:Numerische Mathematik
30208:
30164:
29144:Schmidt decomposition
29020:Canonical correlation
28962:
28894:
28870:canonical multipliers
28824:
28786:
28759:finite-rank operators
28748:
28719:
28677:
28636:
28607:
28576:
28549:
28330:
28303:
28301:{\displaystyle \psi }
28279:
28242:
28205:
28203:{\displaystyle T_{f}}
28174:
28136:
28110:
28086:
28050:
28002:
27964:
27881:
27820:
27818:{\displaystyle U_{2}}
27789:
27787:{\displaystyle U_{1}}
27758:
27724:
27683:
27633:
27583:
27557:
27555:{\displaystyle T_{f}}
27526:
27470:
27446:
27407:
27378:
27346:
27317:
27290:
27268:can be extended to a
27261:
27202:
27173:
27144:
27118:
27079:
27047:
27018:
26989:
26970:for unitary matrices
26963:
26924:
26895:
26852:
26768:
26720:
26677:
26648:
26617:
26540:
26439:
26367:matrices, defined by
26360:
26337:inner product on the
26318:
26282:
26199:
26170:
26110:
26049:
25990:
25920:
25895:
25866:
25832:
25830:{\displaystyle K^{m}}
25801:
25769:
25740:
25709:
25683:
25639:
25613:
25589:Moore–Penrose inverse
25577:
25544:
25511:
25494:corresponding to the
25487:
25454:
25430:
25404:
25380:
25347:
25309:
25279:
25247:
25214:
25181:
25100:
25076:
25038:
25009:
24979:
24948:
24915:
24877:
24847:
24815:
24782:
24749:
24701:
24668:
24642:
24609:
24576:
24552:
24526:
24502:
24429:
24395:
24362:
24312:
24279:
24241:
24211:
24179:
24146:
24113:
24091:singular values from
24084:
24060:
24027:
23998:
23972:
23948:
23915:
23884:
23838:
23765:
23731:
23707:
23681:
23658:distinguished for an
23616:
23601:
22778:
22749:
22725:
22688:
22577:
22530:
22496:
22442:
22393:
22363:
22304:
22229:
22189:
22148:
22107:
22084:solves a sequence of
22073:
22005:
21948:
21922:
21898:
21874:
21802:
21759:
21726:
21694:eigenvalue algorithms
21681:
21648:
21598:
21568:
21530:
21490:
21466:
21437:
21399:
21370:
21325:
21289:
21260:
21221:
21201:
21178:
21158:
21136:
21095:Jacobi transformation
21084:
21048:
21028:
21004:
20984:
20964:
20922:
20902:
20840:
20838:{\displaystyle (p,q)}
20808:
20788:
20768:
20727:
20705:
20628:
20592:
20559:
20533:
20507:
20461:
20323:
20259:
20209:
20169:
20129:
19973:
19841:
19807:
19781:
19748:
19710:
19677:
19644:
19601:
19570:
19534:
19501:
19470:
19430:
19404:
19367:
19341:
19309:
19272:
19235:
19176:
19152:
19109:
19066:
18999:
18970:
18968:{\displaystyle (k-1)}
18936:
18899:
18869:
18840:
18814:
18788:
18759:
18733:
18677:
18639:
18601:
18563:
18525:
18471:
18126:
18100:
18076:
18036:
17931:
17898:
17864:
17797:
17768:
17737:
17526:
17506:
17486:
17464:
17435:
17406:
17374:
17269:
17240:
17206:
17177:
17124:
17091:
16986:
16928:
16877:
16843:
16790:
16731:
16684:
16428:
16347:
16127:
16098:
16069:
16047:
16003:
15867:
15477:
15455:
15421:
15392:
15363:
15296:
15274:
15227:
15193:
15167:
15145:
15125:
15084:
15050:
15030:
15010:
14988:
14947:
14913:
14911:{\displaystyle \ell }
14893:
14867:
14843:
14729:
14705:
14673:
14637:
14609:
14574:
14545:
14518:
14458:
14432:
14403:
14379:
14341:
14315:
14289:
14260:
14234:
14189:
14078:
14050:
14019:
13995:
13975:for some real number
13968:
13887:
13859:
13814:
13793:extreme value theorem
13784:
13677:
13647:
13617:
13591:
13546:
13520:
13430:
13413:: if the rank of the
13407:Schmidt decomposition
13393:
13260:
13215:
13180:
13075:
13046:
13020:
12985:
12930:
12884:
12838:
12787:
12761:
12718:
12677:
12636:
12607:
12578:
12544:
12468:
12442:
12407:
12376:
12343:
12312:
12288:
12255:
12222:
12118:
12092:
12032:
11990:can be written as an
11983:
11939:
11917:
11895:
11862:
11790:
11763:
11701:
11666:
11636:
11610:
11577:
11533:
11509:
11477:
11448:
11411:
11382:
11356:
11330:
11304:
11259:
11235:
11206:under the constraint
11199:
11164:
11127:
11098:
11069:
11036:
10990:
10963:
10933:
10907:
10881:
10855:
10822:
10796:
10770:
10744:
10718:
10689:
10663:
10606:
10584:
10553:
10485:
10434:
10391:
10357:
10328:
10302:
10276:
10250:
10224:
10198:
10172:
10146:
10114:
10068:
10017:
9971:
9935:
9902:
9871:
9837:
9786:
9751:
9725:
9694:
9668:
9642:
9572:
9543:
9502:
9460:
9432:
9391:
9363:
9318:
9288:
9011:
8957:
8920:
8883:
8854:
8828:
8802:
8776:
8750:
8724:
8698:
8672:
8635:
8609:
8583:
8559:
8529:
8503:
8470:
8440:
8414:
8384:
8347:
8308:
8281:
8255:
8229:
8202:
8169:
8129:
8100:
8098:{\displaystyle K^{n}}
8069:
8041:
8012:
8010:{\displaystyle K^{m}}
7981:
7949:
7925:
7899:
7865:
7839:
7813:
7768:
7739:
7713:
7659:
7624:
7598:
7572:
7478:
7476:{\displaystyle K^{n}}
7447:
7421:
7419:{\displaystyle K^{m}}
7390:
7364:
7332:
7295:
7024:
6998:
6636:
6603:
6569:
6536:
6510:
6484:
5881:
5838:
5689:
5652:
5625:
5576:
5574:{\displaystyle T(S).}
5540:
5514:
5512:{\displaystyle T(S).}
5478:
5437:
5404:
5375:
5339:
5306:
5282:
5249:
5216:
5192:
5155:
5115:
5091:
5055:
5028:
5004:
4970:
4946:
4944:{\displaystyle K^{n}}
4915:
4891:
4867:
4865:{\displaystyle K^{m}}
4836:
4834:{\displaystyle K^{n}}
4805:
4753:
4705:
4657:
4628:
4604:
4573:
4463:
4378:linear transformation
4362:
4360:{\displaystyle K^{n}}
4331:
4305:
4251:
4249:{\displaystyle K^{m}}
4216:
4190:
4136:
4110:
4075:
4049:
4023:
3990:
3960:
3927:
3901:
3874:
3837:
3811:
3778:
3748:
3722:
3681:
3655:
3626:
3600:
3576:positive-semidefinite
3567:
3537:
3504:
3476:
3441:
3412:
3383:
3350:
3304:
3271:
3237:
3204:
3181:are orthonormal bases
3169:for further details.
3158:
3130:
3106:
3072:
3045:
3011:
2972:
2936:
2903:
2864:
2835:
2799:
2766:
2733:
2707:
2671:
2638:
2605:
2575:
2545:
2512:
2486:
2456:
2426:
2393:
2361:of three geometrical
2350:
2317:
2283:
2246:
2212:
2179:
2153:
2117:
2086:linear transformation
2077:
2044:
2014:
1981:
1951:
1921:
1892:
1828:
1793:. The SVD decomposes
1765:
1750:fitting of data, and
1715:
1638:
1610:
1583:
1553:
1527:
1498:
1453:
1420:
1394:
1341:
1312:
1286:
1262:
1240:
1206:
1179:
1136:
1068:
1046:
1017:
960:
903:
877:
851:
825:
795:
772:and are known as the
765:
741:
719:
681:The diagonal entries
674:
624:
598:
572:
546:
518:
489:
461:
434:
408:
380:
350:
326:
278:
254:
219:
24:
32317:Proto-value function
32296:Dirichlet eigenvalue
32210:Abstract index group
32095:Approximate identity
32058:Rigged Hilbert space
31934:Krein–Rutman theorem
31780:Involution/*-algebra
31442:Kakutani fixed-point
31427:Riesz representation
30810:Van Loan, Charles F.
30173:
30169:, and remember that
30065:
29694:(20–21): 8930–8945.
29112:Matrix decomposition
29092:Linear least squares
28944:
28878:
28799:
28773:
28732:
28690:
28648:
28620:
28589:
28562:
28342:
28316:
28292:
28255:
28218:
28187:
28149:
28123:
28099:
28075:
28015:
27977:
27901:
27833:
27802:
27771:
27745:
27696:
27648:
27597:
27572:
27539:
27483:
27459:
27420:
27391:
27362:
27330:
27303:
27277:
27223:
27186:
27157:
27131:
27092:
27063:
27031:
27002:
26976:
26937:
26908:
26882:
26795:
26781:
26733:
26704:
26696:Hilbert–Schmidt norm
26661:
26630:
26581:
26553:
26452:
26373:
26343:
26329:Hilbert–Schmidt norm
26293:
26287:(the eigenvalues of
26216:
26183:
26121:
26064:
26003:
25933:
25907:
25877:
25845:
25814:
25788:
25753:
25729:
25696:
25672:
25626:
25597:
25557:
25524:
25500:
25467:
25443:
25417:
25393:
25360:
25322:
25292:
25259:
25227:
25194:
25112:
25089:
25054:
25022:
24992:
24968:
24928:
24890:
24860:
24827:
24795:
24762:
24714:
24681:
24655:
24622:
24589:
24565:
24539:
24515:
24442:
24416:
24379:
24327:
24292:
24254:
24224:
24191:
24159:
24126:
24097:
24073:
24040:
24011:
23985:
23961:
23928:
23895:
23849:
23778:
23752:
23720:
23694:
23664:
22793:
22760:
22736:
22699:
22588:
22560:
22509:
22455:
22406:
22380:
22337:
22242:
22201:
22161:
22135:
22114:eigenvalue methods (
22090:
22056:
21965:
21935:
21911:
21905:is much larger than
21887:
21826:
21772:
21757:{\displaystyle O(n)}
21739:
21724:{\displaystyle O(n)}
21706:
21661:
21618:
21585:
21579:The SVD of a matrix
21543:
21505:
21477:
21453:
21412:
21386:
21345:
21312:
21269:
21230:
21210:
21190:
21167:
21147:
21103:
21073:
21037:
21017:
20993:
20973:
20934:
20911:
20849:
20817:
20797:
20777:
20736:
20716:
20661:
20605:
20572:
20546:
20520:
20477:
20336:
20270:
20220:
20182:
20142:
19986:
19859:
19820:
19794:
19761:
19728:
19690:
19657:
19614:
19583:
19547:
19514:
19483:
19443:
19417:
19380:
19354:
19330:
19285:
19248:
19189:
19165:
19117:
19074:
19008:
18979:
18947:
18912:
18882:
18856:
18827:
18801:
18772:
18746:
18698:
18652:
18614:
18576:
18538:
18484:
18137:
18109:
18087:
18053:
17944:
17911:
17878:
17806:
17777:
17748:
17537:
17515:
17495:
17473:
17444:
17415:
17386:
17348:
17278:
17249:
17215:
17186:
17133:
17104:
17065:
16995:
16937:
16886:
16852:
16799:
16740:
16695:
16441:
16363:
16138:
16107:
16078:
16056:
16014:
15880:
15488:
15464:
15430:
15401:
15372:
15305:
15283:
15236:
15202:
15176:
15154:
15134:
15093:
15059:
15039:
15019:
14997:
14956:
14922:
14902:
14876:
14854:
14740:
14716:
14688:
14648:
14620:
14596:
14561:
14531:
14471:
14445:
14416:
14392:
14354:
14328:
14302:
14273:
14247:
14202:
14091:
14065:
14037:
14008:
13981:
13900:
13874:
13866:Lagrange multipliers
13825:
13801:
13693:
13660:
13634:
13604:
13559:
13533:
13509:
13482:disease surveillance
13417:
13354:
13269:The Kabsch algorithm
13227:
13191:
13088:
13059:
13033:
13007:
12941:
12897:
12851:
12800:
12774:
12731:
12690:
12653:
12620:
12594:
12565:
12481:
12455:
12429:
12389:
12356:
12325:
12301:
12268:
12235:
12131:
12105:
12045:
12039:or, in coordinates,
12000:
11970:
11949:Eckart–Young theorem
11928:
11904:
11873:
11803:
11774:
11712:
11679:
11653:
11625:
11588:
11584:with another matrix
11564:
11520:
11496:
11461:
11428:
11395:
11369:
11343:
11317:
11288:
11246:
11210:
11181:
11151:
11111:
11082:
11049:
11003:
10977:
10950:
10920:
10894:
10868:
10839:
10809:
10783:
10757:
10731:
10702:
10676:
10637:
10617:linear least squares
10593:
10564:
10498:
10447:
10421:
10374:
10341:
10315:
10289:
10263:
10237:
10211:
10185:
10159:
10130:
10081:
10030:
9988:
9948:
9915:
9884:
9850:
9799:
9768:
9738:
9707:
9681:
9675:and diagonal matrix
9655:
9604:
9559:
9515:
9477:
9447:
9404:
9378:
9335:
9305:
9024:
8970:
8944:
8903:
8867:
8841:
8815:
8789:
8763:
8737:
8711:
8685:
8651:
8622:
8596:
8572:
8542:
8516:
8483:
8453:
8427:
8397:
8364:
8356:rank–nullity theorem
8331:
8294:
8268:
8242:
8216:
8182:
8149:
8113:
8082:
8056:
8025:
7994:
7968:
7938:
7912:
7882:
7852:
7826:
7781:
7752:
7726:
7675:
7645:
7611:
7585:
7491:
7460:
7434:
7403:
7377:
7351:
7321:
7037:
7011:
6653:
6616:
6590:
6549:
6523:
6497:
5893:
5851:
5702:
5672:
5638:
5589:
5553:
5527:
5491:
5450:
5435:{\displaystyle T(S)}
5417:
5391:
5352:
5319:
5295:
5262:
5247:{\displaystyle T(S)}
5229:
5205:
5190:{\displaystyle n=m,}
5172:
5132:
5104:
5068:
5044:
5017:
4983:
4977:-th basis vector of
4959:
4928:
4922:-th basis vector of
4904:
4880:
4849:
4818:
4768:
4718:
4670:
4641:
4617:
4586:
4476:
4385:
4344:
4318:
4264:
4233:
4203:
4149:
4123:
4097:
4062:
4036:
4006:
3977:
3940:
3914:
3890:
3854:
3824:
3794:
3765:
3735:
3711:
3668:
3639:
3613:
3587:
3554:
3517:
3488:
3453:
3425:
3396:
3363:
3317:
3284:
3258:
3217:
3191:
3147:
3119:
3089:
3061:
3028:
3000:
2949:
2916:
2877:
2848:
2812:
2779:
2746:
2720:
2684:
2651:
2618:
2588:
2562:
2525:
2499:
2473:
2443:
2413:
2373:
2330:
2296:
2263:
2233:
2227:of the space, while
2192:
2166:
2130:
2093:
2061:
2027:
1994:
1968:
1934:
1908:
1815:
1775:. First, we see the
1652:
1621:
1597:
1566:
1540:
1511:
1466:
1433:
1407:
1360:
1325:
1299:
1273:
1249:
1219:
1190:
1147:
1057:
1028:
973:
919:
890:
864:
838:
812:
782:
752:
728:
685:
639:
611:
585:
559:
533:
498:
472:
448:
417:
395:
363:
337:
289:
265:
237:
202:
16:Matrix decomposition
32418:Functional analysis
32120:Von Neumann algebra
31856:Polar decomposition
31626:Functional calculus
31585:Mahler's conjecture
31564:Von Neumann algebra
31278:Functional analysis
31118:Numerical stability
30932:. Morgan Kaufmann.
30814:Matrix Computations
30782:1965SJNA....2..205G
30702:Trefethen, Lloyd N.
30389:1951PNAS...37..760F
30093:
30033:2022AdSpR..69..620M
29988:2023AsDyn...7..153M
29900:2014arXiv1406.3506F
29842:2013arXiv1304.1467B
29739:2019PhRvD..99b4010S
29626:1992MaCom..58..185E
29560:2013PLoSO...878913B
29501:2006PNAS..10311828A
29495:(32): 11828–11833.
29442:2004PNAS..10116577A
29436:(47): 16577–16582.
29383:2000PNAS...9710101A
29377:(18): 10101–10106.
29175:Wavelet compression
29133:Polar decomposition
28972:valeurs singulières
28921:transformation for
28907:polar decomposition
27893:polar decomposition
26872:Scale-invariant SVD
26796:
26789:
26684:This is called the
26609:
26582:
26575:
25341:
25172:
24909:
24493:
24273:
23829:
23577:
23530:
23483:
23436:
23389:
23342:
23155:
23137:
23119:
23101:
20642:Calculating the SVD
20264:into account gives
20203:
20163:
18454:
18390:
17610:
17556:
17491:is no greater than
17368:
17349:
17342:
17313:
17171:
17085:
17066:
17059:
17030:
16980:
16922:
16837:
16776:
16667:
16613:
16565:
16489:
16315:
16283:
16216:
16161:
15957:
15899:
15780:
15732:
15682:
15634:
15536:
15515:
15055:-th eigenvector of
13467:recommender systems
13403:quantum information
13332:pattern recognition
12532:
12506:
12421:reverse correlation
11171:that minimizes the
11143:total least squares
10248:{\displaystyle {1}}
10170:{\displaystyle {1}}
9980:polar decomposition
8557:{\displaystyle n-m}
8468:{\displaystyle m-n}
8325:, respectively, of
6491:The scaling matrix
4021:{\displaystyle n-r}
3844:are a basis of the
3809:{\displaystyle m-r}
3755:are a basis of the
3246:orthonormal vectors
2257:of each coordinate
1646:semi-unitary matrix
1591:semi-unitary matrix
1127:
858:and the columns of
525:conjugate transpose
228:polar decomposition
154:, another rotation.
32250:Unbounded operator
32179:Essential spectrum
32158:Schur–Horn theorem
32148:Bauer–Fike theorem
32143:Alon–Boppana bound
32136:Finite-Dimensional
32110:Nuclear C*-algebra
31954:Spectral asymmetry
31651:Riemann hypothesis
31350:Topological vector
30924:Samet, H. (2006).
30865:10.1007/BF01937276
30580:10.1007/BF02163027
30479:10.1007/BF02288367
30203:
30159:
30077:
29953:10.3233/IDA-150734
29908:10.1111/exsy.12088
29770:Konstan, Joseph A.
28957:
28934:integral operators
28923:Hermitian matrices
28889:
28819:
28781:
28743:
28714:
28672:
28631:
28602:
28574:{\displaystyle H.}
28571:
28544:
28484:
28398:
28328:{\displaystyle H,}
28325:
28298:
28274:
28237:
28200:
28169:
28143:is compact, so is
28131:
28105:
28081:
28045:
27997:
27959:
27876:
27870:
27815:
27784:
27753:
27719:
27678:
27628:
27578:
27566:multiplication by
27552:
27521:
27465:
27441:
27402:
27373:
27341:
27315:{\displaystyle H.}
27312:
27285:
27256:
27217:The factorization
27197:
27168:
27139:
27113:
27074:
27042:
27013:
26984:
26958:
26919:
26890:
26847:
26811:
26763:
26715:
26672:
26643:
26612:
26595:
26594:
26535:
26434:
26355:
26313:
26277:
26194:
26165:
26105:
26044:
25985:
25915:
25890:
25861:
25827:
25796:
25764:
25735:
25704:
25678:
25634:
25608:
25572:
25539:
25506:
25482:
25449:
25425:
25411:column vectors of
25399:
25375:
25342:
25325:
25304:
25274:
25242:
25209:
25176:
25156:
25095:
25071:
25033:
25004:
24974:
24943:
24910:
24893:
24872:
24842:
24810:
24777:
24744:
24696:
24663:
24637:
24604:
24571:
24547:
24533:column vectors of
24521:
24497:
24477:
24424:
24390:
24357:
24307:
24274:
24257:
24236:
24206:
24174:
24141:
24108:
24079:
24055:
24022:
23993:
23967:
23943:
23910:
23879:
23833:
23813:
23760:
23726:
23702:
23676:
23655:
23596:
23594:
23563:
23516:
23469:
23422:
23375:
23328:
23141:
23123:
23105:
23087:
22773:
22744:
22720:
22683:
22572:
22525:
22491:
22437:
22388:
22358:
22325:Hermitian matrices
22299:
22290:
22224:
22184:
22143:
22102:
22068:
22000:
21943:
21917:
21893:
21869:
21797:
21754:
21721:
21676:
21643:
21593:
21563:
21525:
21485:
21461:
21432:
21394:
21365:
21320:
21298:Numerical approach
21284:
21255:
21216:
21196:
21173:
21153:
21131:
21079:
21043:
21023:
20999:
20979:
20959:
20917:
20897:
20835:
20803:
20783:
20763:
20722:
20700:
20623:
20587:
20554:
20528:
20502:
20456:
20454:
20318:
20254:
20204:
20185:
20164:
20145:
20124:
20122:
19968:
19836:
19802:
19776:
19743:
19705:
19672:
19639:
19596:
19565:
19529:
19496:
19465:
19425:
19399:
19362:
19336:
19304:
19267:
19230:
19171:
19147:
19104:
19061:
18994:
18965:
18931:
18894:
18864:
18835:
18809:
18783:
18754:
18728:
18672:
18634:
18596:
18558:
18520:
18466:
18438:
18400:
18374:
18343:
18292:
18249:
18234:
18176:
18121:
18095:
18071:
18031:
18022:
18007:
17926:
17893:
17859:
17853:
17792:
17763:
17732:
17594:
17540:
17521:
17501:
17481:
17459:
17430:
17401:
17369:
17338:
17288:
17264:
17235:
17201:
17172:
17151:
17119:
17086:
17055:
17005:
16981:
16960:
16923:
16902:
16872:
16838:
16817:
16785:
16760:
16726:
16679:
16651:
16597:
16549:
16473:
16423:
16357:Let us now define
16342:
16340:
16299:
16267:
16200:
16145:
16122:
16093:
16064:
16042:
15998:
15941:
15883:
15874:This implies that
15862:
15853:
15812:
15764:
15716:
15666:
15618:
15601:
15539:
15520:
15499:
15472:
15450:
15416:
15387:
15358:
15352:
15291:
15269:
15222:
15188:
15162:
15140:
15120:
15079:
15045:
15035:-th column is the
15025:
15005:
14983:
14942:
14908:
14888:
14862:
14838:
14829:
14724:
14700:
14682:, there exists an
14668:
14632:
14604:
14569:
14543:{\displaystyle 2n}
14540:
14513:
14453:
14427:
14398:
14374:
14348:its eigenvalue is
14336:
14310:
14284:
14255:
14229:
14184:
14073:
14045:
14014:
14002:The nabla symbol,
13990:
13963:
13882:
13854:
13809:
13779:
13774:
13672:
13642:
13612:
13586:
13541:
13515:
13499:Proof of existence
13425:
13388:
13255:
13210:
13175:
13107:
13070:
13041:
13015:
12980:
12925:
12879:
12833:
12782:
12756:
12713:
12672:
12631:
12602:
12573:
12539:
12518:
12517:
12492:
12463:
12437:
12402:
12371:
12338:
12307:
12283:
12250:
12217:
12176:
12151:
12113:
12087:
12027:
11978:
11934:
11912:
11890:
11857:
11785:
11758:
11696:
11661:
11631:
11605:
11572:
11528:
11504:
11472:
11443:
11406:
11389:span the range of
11377:
11351:
11325:
11299:
11254:
11230:
11194:
11159:
11122:
11093:
11064:
11031:
10985:
10958:
10928:
10902:
10876:
10850:
10817:
10791:
10765:
10739:
10713:
10684:
10658:
10631:can be written as
10601:
10579:
10548:
10480:
10429:
10398:matrix has a SVD.
10386:
10352:
10323:
10297:
10271:
10245:
10219:
10193:
10167:
10141:
10109:
10063:
10012:
9966:
9930:
9897:
9866:
9832:
9781:
9746:
9720:
9689:
9663:
9637:
9567:
9538:
9497:
9455:
9427:
9386:
9358:
9313:
9283:
9281:
9006:
8952:
8915:
8878:
8849:
8823:
8797:
8771:
8745:
8719:
8693:
8667:
8630:
8604:
8578:
8554:
8524:
8498:
8465:
8435:
8409:
8379:
8342:
8303:
8276:
8250:
8224:
8197:
8164:
8124:
8095:
8064:
8036:
8007:
7976:
7944:
7920:
7894:
7860:
7834:
7808:
7763:
7734:
7708:
7654:
7619:
7593:
7567:
7565:
7473:
7442:
7416:
7385:
7359:
7327:
7290:
7284:
7281:
7270:
7259:
7252:
7245:
7229:
7215:
7204:
7197:
7190:
7177:
7166:
7159:
7152:
7145:
7132:
7125:
7118:
7111:
7104:
7095:
7088:
7081:
7074:
7067:
7019:
6993:
6991:
6968:
6778:
6631:
6598:
6564:
6531:
6505:
6479:
6477:
6469:
6466:
6455:
6448:
6441:
6434:
6418:
6411:
6404:
6397:
6390:
6381:
6374:
6367:
6360:
6350:
6341:
6327:
6320:
6313:
6306:
6290:
6283:
6276:
6266:
6259:
6218:
6215:
6204:
6178:
6145:
6108:
6050:
6047:
6040:
6030:
6023:
6014:
6004:
5997:
5990:
5981:
5974:
5967:
5960:
5948:
5941:
5934:
5924:
5876:
5833:
5827:
5684:
5650:{\displaystyle T.}
5647:
5620:
5571:
5535:
5509:
5473:
5432:
5399:
5370:
5334:
5301:
5277:
5244:
5211:
5187:
5150:
5110:
5086:
5050:
5023:
4999:
4965:
4941:
4910:
4886:
4862:
4831:
4800:
4748:
4700:
4652:
4623:
4599:
4568:
4458:
4453:
4373:on these spaces).
4357:
4326:
4300:
4246:
4211:
4185:
4131:
4105:
4070:
4044:
4018:
3997:in the real case).
3985:
3955:
3922:
3896:
3869:
3832:
3806:
3773:
3743:
3717:
3694:eigendecomposition
3676:
3650:
3621:
3595:
3562:
3532:
3499:
3471:
3436:
3407:
3378:
3345:
3299:
3266:
3232:
3199:
3153:
3125:
3101:
3067:
3040:
3006:
2967:
2931:
2898:
2859:
2842:respectively; and
2830:
2794:
2761:
2728:
2702:
2666:
2633:
2600:
2570:
2540:
2507:
2481:
2467:In particular, if
2451:
2421:
2388:
2345:
2312:
2278:
2241:
2207:
2174:
2148:
2112:
2072:
2039:
2009:
1976:
1946:
1916:
1895:
1887:
1823:
1710:
1633:
1605:
1578:
1548:
1522:
1493:
1448:
1415:
1389:
1336:
1307:
1281:
1257:
1235:
1201:
1174:
1131:
1111:
1041:
1012:
955:
898:
872:
846:
820:
790:
760:
736:
714:
669:
619:
593:
567:
541:
513:
484:
456:
429:
403:
375:
345:
321:
273:
249:
214:
190:eigendecomposition
159:
32380:
32379:
32357:Transfer operator
32332:Spectral geometry
32017:Spectral abscissa
31997:Approximate point
31939:Normal eigenvalue
31728:
31727:
31631:Integral operator
31408:
31407:
31244:
31243:
31049:978-0-521-88068-8
30958:978-0-9614088-5-5
30939:978-0-12-369446-1
30916:978-0-521-46713-1
30891:978-0-521-38632-6
30831:GSL Team (2007).
30823:978-0-8018-5414-9
30715:978-0-89871-361-9
30642:978-1-4704-6332-8
30371:Fan, Ky. (1951).
30300:978-0-89871-471-5
29717:Physical Review D
29336:978-1-5386-4658-8
29149:Smith normal form
29007:Christian Reinsch
28981:in 1954–1955 and
28475:
28389:
28084:{\displaystyle T}
27739:. The reason why
27581:{\displaystyle f}
27468:{\displaystyle f}
26842:
26799:
26610:
26585:
26530:
26490:
25982:
25738:{\displaystyle k}
25681:{\displaystyle k}
25509:{\displaystyle t}
25452:{\displaystyle t}
25402:{\displaystyle t}
25126:
25098:{\displaystyle t}
25068:
24977:{\displaystyle r}
24574:{\displaystyle r}
24524:{\displaystyle r}
24082:{\displaystyle k}
23970:{\displaystyle k}
23729:{\displaystyle r}
23590:
23588:
23170:
23168:
21920:{\displaystyle n}
21896:{\displaystyle m}
21607:bidiagonal matrix
21219:{\displaystyle V}
21199:{\displaystyle U}
21176:{\displaystyle J}
21156:{\displaystyle G}
21082:{\displaystyle R}
21046:{\displaystyle M}
21026:{\displaystyle M}
21002:{\displaystyle U}
20982:{\displaystyle V}
20920:{\displaystyle m}
20806:{\displaystyle q}
20786:{\displaystyle p}
20200:
20160:
18435:
18371:
18212:
17985:
17710:
17683:
17658:
17590:
17524:{\displaystyle n}
17504:{\displaystyle m}
16546:
16528:
16470:
16416:
15251:
15143:{\displaystyle j}
15108:
15048:{\displaystyle i}
15028:{\displaystyle i}
14791:
13449:ensemble forecast
13385:
13370:
13328:signal processing
13292:signal processing
13286:Signal processing
13144:
13100:
12845:then the product
12587:orthogonal matrix
12585:to determine the
12534:
12508:
12310:{\displaystyle i}
12167:
12142:
11937:{\displaystyle r}
11887:
11839:
11817:
11739:
11693:
11634:{\displaystyle r}
11602:
9594:using a basis of
8581:{\displaystyle 0}
7947:{\displaystyle p}
7279:
7268:
7243:
7227:
7213:
7188:
7175:
7143:
6645:orthogonal matrix
6582:identity matrices
6464:
6432:
6339:
6304:
6123:
5304:{\displaystyle T}
5214:{\displaystyle T}
5113:{\displaystyle T}
5062:of radius one in
5053:{\displaystyle S}
5026:{\displaystyle T}
4968:{\displaystyle i}
4913:{\displaystyle i}
4889:{\displaystyle T}
4626:{\displaystyle i}
4225:orthonormal basis
4087:Geometric meaning
3899:{\displaystyle r}
3720:{\displaystyle r}
3546:orthonormal bases
3156:{\displaystyle m}
3128:{\displaystyle n}
3070:{\displaystyle n}
3009:{\displaystyle n}
1744:signal processing
912:orthonormal bases
832:. The columns of
133:horizontally and
88:, a rotation, on
33:2 × 2
32425:
32362:Transform theory
32082:Special algebras
32063:Spectral theorem
32026:Spectral Theorem
31866:Spectral theorem
31755:
31748:
31741:
31732:
31731:
31718:
31717:
31636:Jones polynomial
31554:Operator algebra
31298:
31297:
31271:
31264:
31257:
31248:
31247:
31154:Matrix splitting
31092:
31085:
31078:
31069:
31068:
31052:
31030:
31013:
30987:
30962:
30943:
30931:
30920:
30904:
30895:
30876:
30840:
30827:
30801:
30757:
30747:
30719:
30697:
30679:
30646:
30627:
30600:
30599:
30556:
30550:
30543:
30537:
30536:
30497:
30491:
30490:
30457:
30451:
30450:
30448:
30441:
30431:Uhlmann, Jeffrey
30427:
30421:
30420:
30410:
30400:
30368:
30362:
30361:
30343:
30311:
30305:
30304:
30277:"Decompositions"
30272:
30266:
30261:
30255:
30250:
30244:
30239:
30233:
30232:
30220:
30214:
30212:
30210:
30209:
30204:
30168:
30166:
30165:
30160:
30158:
30157:
30148:
30147:
30142:
30136:
30122:
30121:
30116:
30110:
30105:
30104:
30099:
30092:
30087:
30082:
30059:
30053:
30052:
30014:
30008:
30007:
29971:
29965:
29964:
29946:
29926:
29920:
29919:
29893:
29873:
29867:
29866:
29860:
29855:
29853:
29845:
29835:
29825:
29816:
29810:
29809:
29803:
29798:
29796:
29788:
29782:
29765:
29759:
29758:
29732:
29712:
29706:
29705:
29703:
29679:
29673:
29672:
29670:
29655:Linear Alg. Appl
29646:
29640:
29639:
29637:
29620:(197): 185–190.
29611:
29602:
29596:
29595:
29581:
29571:
29539:
29533:
29532:
29522:
29512:
29480:
29474:
29473:
29463:
29453:
29421:
29415:
29414:
29404:
29394:
29362:
29356:
29355:
29353:
29351:
29312:
29306:
29305:
29281:
29275:
29270:
29264:
29263:
29235:
29229:
29228:
29192:
29062:Fourier analysis
28991:Givens rotations
28966:
28964:
28963:
28958:
28956:
28955:
28900:
28898:
28896:
28895:
28890:
28885:
28850:Eugenio Beltrami
28830:
28828:
28826:
28825:
28820:
28818:
28813:
28812:
28807:
28792:
28790:
28788:
28787:
28782:
28780:
28754:
28752:
28750:
28749:
28744:
28739:
28725:
28723:
28721:
28720:
28715:
28710:
28709:
28700:
28683:
28681:
28679:
28678:
28673:
28668:
28667:
28658:
28642:
28640:
28638:
28637:
28632:
28627:
28613:
28611:
28609:
28608:
28603:
28601:
28600:
28582:
28580:
28578:
28577:
28572:
28553:
28551:
28550:
28545:
28540:
28539:
28530:
28525:
28521:
28520:
28519:
28510:
28494:
28493:
28483:
28471:
28470:
28461:
28456:
28452:
28451:
28450:
28441:
28430:
28429:
28424:
28418:
28417:
28408:
28397:
28382:
28381:
28376:
28370:
28369:
28360:
28349:
28336:
28334:
28332:
28331:
28326:
28309:
28307:
28305:
28304:
28299:
28285:
28283:
28281:
28280:
28275:
28270:
28269:
28248:
28246:
28244:
28243:
28238:
28233:
28232:
28211:
28209:
28207:
28206:
28201:
28199:
28198:
28180:
28178:
28176:
28175:
28170:
28168:
28163:
28162:
28157:
28142:
28140:
28138:
28137:
28132:
28130:
28116:
28114:
28112:
28111:
28106:
28092:
28090:
28088:
28087:
28082:
28056:
28054:
28052:
28051:
28046:
28044:
28043:
28038:
28032:
28031:
28022:
28008:
28006:
28004:
28003:
27998:
27996:
27995:
27990:
27984:
27971:and notice that
27968:
27966:
27965:
27960:
27958:
27957:
27952:
27946:
27945:
27936:
27928:
27927:
27922:
27916:
27908:
27885:
27883:
27882:
27877:
27875:
27874:
27867:
27866:
27853:
27852:
27826:
27824:
27822:
27821:
27816:
27814:
27813:
27795:
27793:
27791:
27790:
27785:
27783:
27782:
27764:
27762:
27760:
27759:
27754:
27752:
27731:as given by the
27730:
27728:
27726:
27725:
27720:
27715:
27710:
27709:
27704:
27689:
27687:
27685:
27684:
27679:
27677:
27676:
27671:
27665:
27664:
27655:
27639:
27637:
27635:
27634:
27629:
27609:
27608:
27589:
27587:
27585:
27584:
27579:
27563:
27561:
27559:
27558:
27553:
27551:
27550:
27530:
27528:
27527:
27522:
27520:
27519:
27514:
27508:
27507:
27498:
27490:
27476:
27474:
27472:
27471:
27466:
27452:
27450:
27448:
27447:
27442:
27414:a measure space
27413:
27411:
27409:
27408:
27403:
27398:
27384:
27382:
27380:
27379:
27374:
27369:
27355:partial isometry
27352:
27350:
27348:
27347:
27342:
27337:
27323:
27321:
27319:
27318:
27313:
27296:
27294:
27292:
27291:
27286:
27284:
27270:bounded operator
27267:
27265:
27263:
27262:
27257:
27255:
27254:
27249:
27243:
27238:
27230:
27208:
27206:
27204:
27203:
27198:
27193:
27179:
27177:
27175:
27174:
27169:
27164:
27150:
27148:
27146:
27145:
27140:
27138:
27124:
27122:
27120:
27119:
27114:
27109:
27104:
27099:
27085:
27083:
27081:
27080:
27075:
27070:
27053:
27051:
27049:
27048:
27043:
27038:
27024:
27022:
27020:
27019:
27014:
27009:
26995:
26993:
26991:
26990:
26985:
26983:
26969:
26967:
26965:
26964:
26959:
26954:
26949:
26944:
26930:
26928:
26926:
26925:
26920:
26915:
26901:
26899:
26897:
26896:
26891:
26889:
26856:
26854:
26853:
26848:
26843:
26841:
26840:
26835:
26829:
26828:
26816:
26810:
26798:
26797:
26794:
26785:
26775:coincides with:
26774:
26772:
26770:
26769:
26764:
26759:
26758:
26740:
26726:
26724:
26722:
26721:
26716:
26711:
26683:
26681:
26679:
26678:
26673:
26668:
26654:
26652:
26650:
26649:
26644:
26642:
26641:
26621:
26619:
26618:
26613:
26611:
26608:
26603:
26593:
26584:
26583:
26580:
26571:
26563:
26544:
26542:
26541:
26536:
26531:
26529:
26525:
26524:
26519:
26518:
26513:
26496:
26491:
26486:
26478:
26470:
26462:
26443:
26441:
26440:
26435:
26430:
26426:
26425:
26420:
26419:
26414:
26391:
26383:
26366:
26364:
26362:
26361:
26356:
26324:
26322:
26320:
26319:
26314:
26312:
26307:
26306:
26301:
26286:
26284:
26283:
26278:
26276:
26275:
26271:
26258:
26253:
26252:
26247:
26226:
26205:
26203:
26201:
26200:
26195:
26190:
26176:
26174:
26172:
26171:
26166:
26161:
26160:
26156:
26143:
26138:
26137:
26132:
26114:
26112:
26111:
26106:
26104:
26103:
26099:
26086:
26081:
26080:
26075:
26055:
26053:
26051:
26050:
26045:
26043:
26042:
26038:
26025:
26020:
26019:
26014:
25994:
25992:
25991:
25986:
25984:
25983:
25975:
25969:
25964:
25963:
25958:
25943:
25926:
25924:
25922:
25921:
25916:
25914:
25899:
25897:
25896:
25891:
25889:
25888:
25872:
25870:
25868:
25867:
25862:
25857:
25856:
25838:
25836:
25834:
25833:
25828:
25826:
25825:
25807:
25805:
25803:
25802:
25797:
25795:
25775:
25773:
25771:
25770:
25765:
25760:
25746:
25744:
25742:
25741:
25736:
25715:
25713:
25711:
25710:
25705:
25703:
25689:
25687:
25685:
25684:
25679:
25645:
25643:
25641:
25640:
25635:
25633:
25619:
25617:
25615:
25614:
25609:
25604:
25583:
25581:
25579:
25578:
25573:
25550:
25548:
25546:
25545:
25540:
25538:
25537:
25532:
25517:
25515:
25513:
25512:
25507:
25493:
25491:
25489:
25488:
25483:
25481:
25480:
25475:
25460:
25458:
25456:
25455:
25450:
25436:
25434:
25432:
25431:
25426:
25424:
25410:
25408:
25406:
25405:
25400:
25386:
25384:
25382:
25381:
25376:
25353:
25351:
25349:
25348:
25343:
25340:
25335:
25330:
25315:
25313:
25311:
25310:
25305:
25285:
25283:
25281:
25280:
25275:
25273:
25272:
25267:
25253:
25251:
25249:
25248:
25243:
25220:
25218:
25216:
25215:
25210:
25208:
25207:
25202:
25185:
25183:
25182:
25177:
25171:
25166:
25161:
25155:
25154:
25149:
25143:
25142:
25137:
25128:
25127:
25122:
25117:
25106:
25104:
25102:
25101:
25096:
25082:
25080:
25078:
25077:
25072:
25070:
25069:
25064:
25059:
25044:
25042:
25040:
25039:
25034:
25029:
25015:
25013:
25011:
25010:
25005:
24985:
24983:
24981:
24980:
24975:
24954:
24952:
24950:
24949:
24944:
24921:
24919:
24917:
24916:
24911:
24908:
24903:
24898:
24883:
24881:
24879:
24878:
24873:
24853:
24851:
24849:
24848:
24843:
24841:
24840:
24835:
24821:
24819:
24817:
24816:
24811:
24788:
24786:
24784:
24783:
24778:
24776:
24775:
24770:
24755:
24753:
24751:
24750:
24745:
24707:
24705:
24703:
24702:
24697:
24695:
24694:
24689:
24674:
24672:
24670:
24669:
24664:
24662:
24648:
24646:
24644:
24643:
24638:
24636:
24635:
24630:
24615:
24613:
24611:
24610:
24605:
24603:
24602:
24597:
24582:
24580:
24578:
24577:
24572:
24558:
24556:
24554:
24553:
24548:
24546:
24532:
24530:
24528:
24527:
24522:
24506:
24504:
24503:
24498:
24492:
24487:
24482:
24476:
24475:
24470:
24464:
24463:
24458:
24449:
24435:
24433:
24431:
24430:
24425:
24423:
24401:
24399:
24397:
24396:
24391:
24386:
24371:QR decomposition
24368:
24366:
24364:
24363:
24358:
24318:
24316:
24314:
24313:
24308:
24285:
24283:
24281:
24280:
24275:
24272:
24267:
24262:
24247:
24245:
24243:
24242:
24237:
24217:
24215:
24213:
24212:
24207:
24205:
24204:
24199:
24185:
24183:
24181:
24180:
24175:
24152:
24150:
24148:
24147:
24142:
24140:
24139:
24134:
24119:
24117:
24115:
24114:
24109:
24104:
24090:
24088:
24086:
24085:
24080:
24066:
24064:
24062:
24061:
24056:
24054:
24053:
24048:
24033:
24031:
24029:
24028:
24023:
24018:
24004:
24002:
24000:
23999:
23994:
23992:
23978:
23976:
23974:
23973:
23968:
23954:
23952:
23950:
23949:
23944:
23942:
23941:
23936:
23921:
23919:
23917:
23916:
23911:
23909:
23908:
23903:
23888:
23886:
23885:
23880:
23842:
23840:
23839:
23834:
23828:
23823:
23818:
23812:
23811:
23806:
23800:
23799:
23794:
23785:
23771:
23769:
23767:
23766:
23761:
23759:
23737:
23735:
23733:
23732:
23727:
23713:
23711:
23709:
23708:
23703:
23701:
23687:
23685:
23683:
23682:
23677:
23652:
23646:
23640:
23634:
23628:
23622:
23605:
23603:
23602:
23597:
23595:
23591:
23589:
23587:
23586:
23576:
23571:
23562:
23561:
23540:
23539:
23529:
23524:
23515:
23514:
23493:
23492:
23482:
23477:
23468:
23467:
23446:
23445:
23435:
23430:
23421:
23420:
23399:
23398:
23388:
23383:
23374:
23373:
23352:
23351:
23341:
23336:
23327:
23326:
23308:
23300:
23299:
23294:
23288:
23287:
23278:
23270:
23269:
23264:
23258:
23257:
23248:
23240:
23239:
23234:
23228:
23227:
23218:
23210:
23209:
23204:
23198:
23197:
23188:
23183:
23175:
23171:
23169:
23167:
23166:
23161:
23154:
23149:
23136:
23131:
23118:
23113:
23100:
23095:
23086:
23078:
23077:
23072:
23071:
23064:
23063:
23058:
23052:
23051:
23042:
23034:
23033:
23028:
23022:
23021:
23012:
23004:
23003:
22998:
22992:
22991:
22982:
22974:
22973:
22968:
22962:
22961:
22952:
22947:
22946:
22940:
22935:
22934:
22929:
22923:
22922:
22913:
22905:
22904:
22899:
22893:
22892:
22883:
22875:
22874:
22869:
22863:
22862:
22853:
22845:
22844:
22839:
22833:
22832:
22823:
22818:
22809:
22808:
22782:
22780:
22779:
22774:
22772:
22771:
22755:
22753:
22751:
22750:
22745:
22743:
22729:
22727:
22726:
22721:
22719:
22711:
22710:
22692:
22690:
22689:
22684:
22682:
22681:
22672:
22671:
22659:
22658:
22649:
22648:
22636:
22635:
22626:
22625:
22613:
22608:
22607:
22595:
22583:
22581:
22579:
22578:
22573:
22543:LQ decomposition
22539:QR decomposition
22536:
22534:
22532:
22531:
22526:
22524:
22516:
22502:
22500:
22498:
22497:
22492:
22487:
22486:
22481:
22475:
22470:
22462:
22448:
22446:
22444:
22443:
22438:
22433:
22432:
22427:
22421:
22413:
22399:
22397:
22395:
22394:
22389:
22387:
22372:LQ decomposition
22369:
22367:
22365:
22364:
22359:
22357:
22352:
22344:
22329:QR decomposition
22321:LQ decomposition
22317:QR decomposition
22308:
22306:
22305:
22300:
22295:
22294:
22287:
22280:
22279:
22274:
22264:
22257:
22235:
22233:
22231:
22230:
22225:
22220:
22215:
22214:
22209:
22195:
22193:
22191:
22190:
22185:
22180:
22179:
22174:
22168:
22154:
22152:
22150:
22149:
22144:
22142:
22113:
22111:
22109:
22108:
22103:
22079:
22077:
22075:
22074:
22069:
22011:
22009:
22007:
22006:
22001:
21999:
21998:
21983:
21982:
21957:QR decomposition
21954:
21952:
21950:
21949:
21944:
21942:
21928:
21926:
21924:
21923:
21918:
21904:
21902:
21900:
21899:
21894:
21880:
21878:
21876:
21875:
21870:
21865:
21860:
21859:
21844:
21843:
21808:
21806:
21804:
21803:
21798:
21793:
21792:
21765:
21763:
21761:
21760:
21755:
21732:
21730:
21728:
21727:
21722:
21690:iterative method
21687:
21685:
21683:
21682:
21677:
21654:
21652:
21650:
21649:
21644:
21639:
21638:
21604:
21602:
21600:
21599:
21594:
21592:
21574:
21572:
21570:
21569:
21564:
21562:
21561:
21556:
21550:
21536:
21534:
21532:
21531:
21526:
21524:
21519:
21518:
21513:
21494:
21492:
21491:
21486:
21484:
21472:
21470:
21468:
21467:
21462:
21460:
21443:
21441:
21439:
21438:
21433:
21431:
21426:
21425:
21420:
21405:
21403:
21401:
21400:
21395:
21393:
21376:
21374:
21372:
21371:
21366:
21364:
21363:
21358:
21352:
21331:
21329:
21327:
21326:
21321:
21319:
21293:
21291:
21290:
21285:
21264:
21262:
21261:
21256:
21251:
21250:
21225:
21223:
21222:
21217:
21205:
21203:
21202:
21197:
21182:
21180:
21179:
21174:
21162:
21160:
21159:
21154:
21140:
21138:
21137:
21132:
21121:
21120:
21088:
21086:
21085:
21080:
21067:QR decomposition
21052:
21050:
21049:
21044:
21032:
21030:
21029:
21024:
21008:
21006:
21005:
21000:
20988:
20986:
20985:
20980:
20968:
20966:
20965:
20960:
20958:
20957:
20926:
20924:
20923:
20918:
20906:
20904:
20903:
20898:
20844:
20842:
20841:
20836:
20812:
20810:
20809:
20804:
20792:
20790:
20789:
20784:
20772:
20770:
20769:
20764:
20731:
20729:
20728:
20723:
20712:where the angle
20709:
20707:
20706:
20701:
20634:
20632:
20630:
20629:
20624:
20619:
20618:
20613:
20598:
20596:
20594:
20593:
20588:
20586:
20585:
20580:
20565:
20563:
20561:
20560:
20555:
20553:
20539:
20537:
20535:
20534:
20529:
20527:
20514:over normalized
20513:
20511:
20509:
20508:
20503:
20498:
20490:
20465:
20463:
20462:
20457:
20455:
20448:
20447:
20442:
20436:
20435:
20419:
20418:
20413:
20407:
20406:
20401:
20388:
20387:
20382:
20376:
20375:
20359:
20358:
20353:
20347:
20327:
20325:
20324:
20319:
20314:
20313:
20298:
20297:
20282:
20281:
20263:
20261:
20260:
20255:
20244:
20230:
20215:
20213:
20211:
20210:
20205:
20202:
20201:
20198:
20195:
20190:
20175:
20173:
20171:
20170:
20165:
20162:
20161:
20158:
20155:
20150:
20133:
20131:
20130:
20125:
20123:
20116:
20115:
20110:
20104:
20103:
20078:
20077:
20072:
20066:
20065:
20060:
20041:
20040:
20035:
20029:
20028:
20009:
20008:
20003:
19997:
19977:
19975:
19974:
19969:
19967:
19962:
19961:
19956:
19944:
19943:
19931:
19926:
19925:
19920:
19908:
19907:
19895:
19890:
19885:
19884:
19879:
19847:
19845:
19843:
19842:
19837:
19832:
19831:
19813:
19811:
19809:
19808:
19803:
19801:
19787:
19785:
19783:
19782:
19777:
19775:
19774:
19769:
19754:
19752:
19750:
19749:
19744:
19742:
19741:
19736:
19716:
19714:
19712:
19711:
19706:
19704:
19703:
19698:
19683:
19681:
19679:
19678:
19673:
19671:
19670:
19665:
19650:
19648:
19646:
19645:
19640:
19635:
19627:
19607:
19605:
19603:
19602:
19597:
19595:
19594:
19576:
19574:
19572:
19571:
19566:
19561:
19560:
19555:
19540:
19538:
19536:
19535:
19530:
19528:
19527:
19522:
19507:
19505:
19503:
19502:
19497:
19495:
19494:
19476:
19474:
19472:
19471:
19466:
19461:
19460:
19436:
19434:
19432:
19431:
19426:
19424:
19410:
19408:
19406:
19405:
19400:
19398:
19397:
19373:
19371:
19369:
19368:
19363:
19361:
19347:
19345:
19343:
19342:
19337:
19315:
19313:
19311:
19310:
19305:
19303:
19302:
19278:
19276:
19274:
19273:
19268:
19266:
19265:
19241:
19239:
19237:
19236:
19231:
19226:
19225:
19207:
19206:
19182:
19180:
19178:
19177:
19172:
19156:
19154:
19153:
19148:
19143:
19142:
19124:
19113:
19111:
19110:
19105:
19100:
19099:
19081:
19070:
19068:
19067:
19062:
19057:
19052:
19047:
19046:
19041:
19029:
19021:
19003:
19001:
19000:
18995:
18993:
18992:
18987:
18974:
18972:
18971:
18966:
18942:
18940:
18938:
18937:
18932:
18930:
18929:
18905:
18903:
18901:
18900:
18895:
18875:
18873:
18871:
18870:
18865:
18863:
18846:
18844:
18842:
18841:
18836:
18834:
18820:
18818:
18816:
18815:
18810:
18808:
18794:
18792:
18790:
18789:
18784:
18779:
18765:
18763:
18761:
18760:
18755:
18753:
18739:
18737:
18735:
18734:
18729:
18724:
18719:
18714:
18713:
18712:
18706:
18683:
18681:
18679:
18678:
18673:
18671:
18666:
18665:
18660:
18645:
18643:
18641:
18640:
18635:
18633:
18632:
18627:
18621:
18607:
18605:
18603:
18602:
18597:
18595:
18590:
18589:
18584:
18569:
18567:
18565:
18564:
18559:
18557:
18556:
18551:
18545:
18529:
18527:
18526:
18521:
18516:
18515:
18510:
18504:
18499:
18491:
18475:
18473:
18472:
18467:
18462:
18453:
18448:
18443:
18437:
18436:
18428:
18426:
18420:
18419:
18414:
18405:
18404:
18389:
18384:
18379:
18373:
18372:
18364:
18362:
18348:
18347:
18340:
18339:
18334:
18326:
18325:
18320:
18303:
18302:
18297:
18296:
18289:
18288:
18283:
18275:
18274:
18269:
18254:
18253:
18239:
18238:
18214:
18213:
18205:
18199:
18181:
18180:
18173:
18172:
18167:
18159:
18158:
18153:
18130:
18128:
18127:
18122:
18104:
18102:
18101:
18096:
18094:
18082:
18080:
18078:
18077:
18072:
18067:
18066:
18061:
18040:
18038:
18037:
18032:
18027:
18026:
18012:
18011:
17987:
17986:
17978:
17976:
17951:
17937:
17935:
17933:
17932:
17927:
17925:
17924:
17919:
17905:we already have
17904:
17902:
17900:
17899:
17894:
17892:
17891:
17886:
17868:
17866:
17865:
17860:
17858:
17857:
17850:
17849:
17844:
17836:
17835:
17830:
17813:
17801:
17799:
17798:
17793:
17791:
17790:
17785:
17772:
17770:
17769:
17764:
17762:
17761:
17756:
17741:
17739:
17738:
17733:
17728:
17727:
17726:
17713:
17712:
17711:
17703:
17697:
17691:
17686:
17685:
17684:
17676:
17670:
17661:
17660:
17659:
17651:
17645:
17639:
17638:
17633:
17627:
17622:
17621:
17616:
17609:
17604:
17599:
17593:
17592:
17591:
17583:
17577:
17568:
17567:
17562:
17555:
17550:
17545:
17530:
17528:
17527:
17522:
17510:
17508:
17507:
17502:
17490:
17488:
17487:
17482:
17480:
17468:
17466:
17465:
17460:
17458:
17457:
17452:
17439:
17437:
17436:
17431:
17429:
17428:
17423:
17410:
17408:
17407:
17402:
17400:
17399:
17394:
17378:
17376:
17375:
17370:
17367:
17362:
17351:
17350:
17347:
17337:
17336:
17330:
17329:
17324:
17318:
17312:
17308:
17296:
17287:
17286:
17273:
17271:
17270:
17265:
17263:
17262:
17257:
17244:
17242:
17241:
17236:
17234:
17229:
17228:
17223:
17210:
17208:
17207:
17202:
17200:
17199:
17194:
17181:
17179:
17178:
17173:
17170:
17165:
17150:
17149:
17144:
17128:
17126:
17125:
17120:
17118:
17117:
17112:
17095:
17093:
17092:
17087:
17084:
17079:
17068:
17067:
17064:
17054:
17053:
17047:
17046:
17041:
17035:
17029:
17025:
17013:
17004:
17003:
16990:
16988:
16987:
16982:
16979:
16974:
16959:
16958:
16953:
16947:
16932:
16930:
16929:
16924:
16921:
16916:
16901:
16900:
16881:
16879:
16878:
16873:
16871:
16866:
16865:
16860:
16847:
16845:
16844:
16839:
16836:
16831:
16816:
16815:
16810:
16794:
16792:
16791:
16786:
16784:
16775:
16770:
16765:
16759:
16758:
16753:
16747:
16735:
16733:
16732:
16727:
16722:
16714:
16713:
16708:
16702:
16688:
16686:
16685:
16680:
16675:
16666:
16661:
16656:
16647:
16646:
16641:
16635:
16624:
16612:
16607:
16602:
16596:
16595:
16590:
16581:
16573:
16564:
16559:
16554:
16548:
16547:
16539:
16537:
16531:
16530:
16529:
16521:
16515:
16509:
16508:
16503:
16497:
16488:
16483:
16478:
16472:
16471:
16463:
16461:
16455:
16454:
16449:
16432:
16430:
16429:
16424:
16419:
16418:
16417:
16409:
16403:
16397:
16396:
16391:
16385:
16377:
16376:
16371:
16351:
16349:
16348:
16343:
16341:
16334:
16333:
16328:
16314:
16309:
16304:
16298:
16297:
16292:
16282:
16277:
16272:
16266:
16265:
16260:
16247:
16246:
16241:
16228:
16227:
16222:
16215:
16210:
16205:
16192:
16191:
16186:
16173:
16172:
16167:
16160:
16155:
16150:
16131:
16129:
16128:
16123:
16121:
16120:
16115:
16102:
16100:
16099:
16094:
16092:
16091:
16086:
16073:
16071:
16070:
16065:
16063:
16051:
16049:
16048:
16043:
16041:
16033:
16032:
16027:
16021:
16007:
16005:
16004:
15999:
15994:
15986:
15985:
15980:
15974:
15969:
15968:
15963:
15956:
15951:
15946:
15936:
15928:
15927:
15922:
15916:
15911:
15910:
15905:
15898:
15893:
15888:
15871:
15869:
15868:
15863:
15858:
15857:
15833:
15817:
15816:
15809:
15808:
15803:
15797:
15792:
15791:
15786:
15779:
15774:
15769:
15761:
15760:
15755:
15749:
15744:
15743:
15738:
15731:
15726:
15721:
15711:
15710:
15705:
15699:
15694:
15693:
15688:
15681:
15676:
15671:
15663:
15662:
15657:
15651:
15646:
15645:
15640:
15633:
15628:
15623:
15606:
15605:
15598:
15597:
15592:
15582:
15581:
15576:
15561:
15556:
15555:
15550:
15544:
15543:
15535:
15530:
15525:
15514:
15509:
15504:
15481:
15479:
15478:
15473:
15471:
15459:
15457:
15456:
15451:
15449:
15444:
15443:
15438:
15425:
15423:
15422:
15417:
15415:
15414:
15409:
15396:
15394:
15393:
15388:
15386:
15385:
15380:
15367:
15365:
15364:
15359:
15357:
15356:
15349:
15348:
15343:
15335:
15334:
15329:
15312:
15300:
15298:
15297:
15292:
15290:
15278:
15276:
15275:
15270:
15262:
15261:
15253:
15252:
15247:
15242:
15232:with eigenvalue
15231:
15229:
15228:
15223:
15221:
15216:
15215:
15210:
15197:
15195:
15194:
15189:
15171:
15169:
15168:
15163:
15161:
15149:
15147:
15146:
15141:
15130:. Moreover, the
15129:
15127:
15126:
15121:
15119:
15118:
15110:
15109:
15104:
15099:
15088:
15086:
15085:
15080:
15078:
15073:
15072:
15067:
15054:
15052:
15051:
15046:
15034:
15032:
15031:
15026:
15014:
15012:
15011:
15006:
15004:
14992:
14990:
14989:
14984:
14951:
14949:
14948:
14943:
14941:
14936:
14935:
14930:
14917:
14915:
14914:
14909:
14897:
14895:
14894:
14889:
14871:
14869:
14868:
14863:
14861:
14847:
14845:
14844:
14839:
14834:
14833:
14809:
14793:
14792:
14787:
14782:
14776:
14771:
14766:
14765:
14760:
14754:
14753:
14748:
14733:
14731:
14730:
14725:
14723:
14711:
14709:
14707:
14706:
14701:
14680:spectral theorem
14677:
14675:
14674:
14669:
14667:
14662:
14661:
14656:
14643:
14641:
14639:
14638:
14633:
14613:
14611:
14610:
14605:
14603:
14580:
14578:
14576:
14575:
14570:
14568:
14552:real variables.
14551:
14549:
14547:
14546:
14541:
14524:
14522:
14520:
14519:
14514:
14512:
14507:
14502:
14501:
14496:
14484:
14464:
14462:
14460:
14459:
14454:
14452:
14438:
14436:
14434:
14433:
14428:
14423:
14409:
14407:
14405:
14404:
14399:
14385:
14383:
14381:
14380:
14375:
14367:
14347:
14345:
14343:
14342:
14337:
14335:
14321:
14319:
14317:
14316:
14311:
14309:
14295:
14293:
14291:
14290:
14285:
14280:
14266:
14264:
14262:
14261:
14256:
14254:
14240:
14238:
14236:
14235:
14230:
14225:
14214:
14209:
14193:
14191:
14190:
14185:
14180:
14172:
14161:
14147:
14142:
14141:
14136:
14118:
14113:
14108:
14107:
14102:
14084:
14082:
14080:
14079:
14074:
14072:
14058:
14056:
14054:
14052:
14051:
14046:
14044:
14025:
14023:
14021:
14020:
14015:
14001:
13999:
13997:
13996:
13991:
13972:
13970:
13969:
13964:
13956:
13951:
13950:
13945:
13927:
13922:
13917:
13916:
13911:
13893:
13891:
13889:
13888:
13883:
13881:
13863:
13861:
13860:
13855:
13838:
13820:
13818:
13816:
13815:
13810:
13808:
13788:
13786:
13785:
13780:
13778:
13775:
13771:
13766:
13761:
13760:
13755:
13742:
13733:
13721:
13720:
13715:
13685:symmetric matrix
13683:
13681:
13679:
13678:
13673:
13653:
13651:
13649:
13648:
13643:
13641:
13623:
13621:
13619:
13618:
13613:
13611:
13597:
13595:
13593:
13592:
13587:
13582:
13571:
13566:
13552:
13550:
13548:
13547:
13542:
13540:
13526:
13524:
13522:
13521:
13516:
13478:higher-order SVD
13434:
13432:
13431:
13426:
13424:
13397:
13395:
13394:
13389:
13387:
13386:
13383:
13377:
13372:
13371:
13368:
13312:inverse problems
13296:image processing
13275:Kabsch algorithm
13264:
13262:
13261:
13256:
13254:
13249:
13248:
13243:
13234:
13219:
13217:
13216:
13211:
13209:
13208:
13184:
13182:
13181:
13176:
13171:
13163:
13158:
13157:
13152:
13145:
13142:
13139:
13138:
13129:
13121:
13116:
13108:
13095:
13081:
13079:
13077:
13076:
13071:
13066:
13052:
13050:
13048:
13047:
13042:
13040:
13026:
13024:
13022:
13021:
13016:
13014:
12989:
12987:
12986:
12981:
12979:
12974:
12970:
12961:
12956:
12948:
12936:
12934:
12932:
12931:
12926:
12924:
12923:
12918:
12912:
12904:
12890:
12888:
12886:
12885:
12880:
12878:
12877:
12872:
12866:
12858:
12844:
12842:
12840:
12839:
12834:
12832:
12831:
12826:
12820:
12815:
12807:
12793:
12791:
12789:
12788:
12783:
12781:
12767:
12765:
12763:
12762:
12757:
12755:
12754:
12749:
12743:
12738:
12724:
12722:
12720:
12719:
12714:
12709:
12708:
12703:
12697:
12683:
12681:
12679:
12678:
12673:
12668:
12660:
12642:
12640:
12638:
12637:
12632:
12627:
12613:
12611:
12609:
12608:
12603:
12601:
12584:
12582:
12580:
12579:
12574:
12572:
12548:
12546:
12545:
12540:
12535:
12533:
12531:
12526:
12516:
12505:
12500:
12491:
12474:
12472:
12470:
12469:
12464:
12462:
12448:
12446:
12444:
12443:
12438:
12436:
12413:
12411:
12409:
12408:
12403:
12401:
12400:
12382:
12380:
12378:
12377:
12372:
12370:
12369:
12364:
12349:
12347:
12345:
12344:
12339:
12337:
12336:
12318:
12316:
12314:
12313:
12308:
12294:
12292:
12290:
12289:
12284:
12282:
12281:
12276:
12261:
12259:
12257:
12256:
12251:
12249:
12248:
12243:
12226:
12224:
12223:
12218:
12213:
12212:
12207:
12198:
12197:
12192:
12186:
12185:
12175:
12163:
12162:
12157:
12150:
12138:
12124:
12122:
12120:
12119:
12114:
12112:
12098:
12096:
12094:
12093:
12088:
12083:
12082:
12073:
12072:
12060:
12059:
12038:
12036:
12034:
12033:
12028:
12023:
12015:
12007:
11989:
11987:
11985:
11984:
11979:
11977:
11960:Separable models
11945:
11943:
11941:
11940:
11935:
11921:
11919:
11918:
11913:
11911:
11899:
11897:
11896:
11891:
11889:
11888:
11883:
11878:
11866:
11864:
11863:
11858:
11853:
11852:
11847:
11841:
11840:
11835:
11830:
11827:
11819:
11818:
11813:
11808:
11796:
11794:
11792:
11791:
11786:
11781:
11767:
11765:
11764:
11759:
11748:
11747:
11741:
11740:
11735:
11730:
11727:
11726:
11707:
11705:
11703:
11702:
11697:
11695:
11694:
11689:
11684:
11672:
11670:
11668:
11667:
11662:
11660:
11642:
11640:
11638:
11637:
11632:
11614:
11612:
11611:
11606:
11604:
11603:
11598:
11593:
11583:
11581:
11579:
11578:
11573:
11571:
11542:of a matrix, as
11537:
11535:
11534:
11529:
11527:
11515:
11513:
11511:
11510:
11505:
11503:
11483:
11481:
11479:
11478:
11473:
11468:
11454:
11452:
11450:
11449:
11444:
11442:
11441:
11436:
11417:
11415:
11413:
11412:
11407:
11402:
11388:
11386:
11384:
11383:
11378:
11376:
11362:
11360:
11358:
11357:
11352:
11350:
11336:
11334:
11332:
11331:
11326:
11324:
11310:
11308:
11306:
11305:
11300:
11295:
11265:
11263:
11261:
11260:
11255:
11253:
11239:
11237:
11236:
11231:
11220:
11205:
11203:
11201:
11200:
11195:
11193:
11188:
11170:
11168:
11166:
11165:
11160:
11158:
11133:
11131:
11129:
11128:
11123:
11118:
11104:
11102:
11100:
11099:
11094:
11089:
11075:
11073:
11071:
11070:
11065:
11063:
11062:
11057:
11042:
11040:
11038:
11037:
11032:
11030:
11022:
11017:
11016:
11011:
10996:
10994:
10992:
10991:
10986:
10984:
10969:
10967:
10965:
10964:
10959:
10957:
10939:
10937:
10935:
10934:
10929:
10927:
10913:
10911:
10909:
10908:
10903:
10901:
10887:
10885:
10883:
10882:
10877:
10875:
10861:
10859:
10857:
10856:
10851:
10846:
10828:
10826:
10824:
10823:
10818:
10816:
10802:
10800:
10798:
10797:
10792:
10790:
10776:
10774:
10772:
10771:
10766:
10764:
10750:
10748:
10746:
10745:
10740:
10738:
10724:
10722:
10720:
10719:
10714:
10709:
10695:
10693:
10691:
10690:
10685:
10683:
10669:
10667:
10665:
10664:
10659:
10657:
10649:
10644:
10610:
10608:
10607:
10602:
10600:
10588:
10586:
10585:
10580:
10578:
10577:
10572:
10557:
10555:
10554:
10549:
10544:
10543:
10538:
10532:
10531:
10526:
10520:
10512:
10511:
10506:
10491:
10489:
10487:
10486:
10481:
10479:
10478:
10473:
10467:
10462:
10454:
10440:
10438:
10436:
10435:
10430:
10428:
10397:
10395:
10393:
10392:
10387:
10363:
10361:
10359:
10358:
10353:
10348:
10334:
10332:
10330:
10329:
10324:
10322:
10308:
10306:
10304:
10303:
10298:
10296:
10282:
10280:
10278:
10277:
10272:
10270:
10256:
10254:
10252:
10251:
10246:
10244:
10230:
10228:
10226:
10225:
10220:
10218:
10204:
10202:
10200:
10199:
10194:
10192:
10178:
10176:
10174:
10173:
10168:
10166:
10152:
10150:
10148:
10147:
10142:
10137:
10120:
10118:
10116:
10115:
10110:
10108:
10107:
10102:
10096:
10088:
10074:
10072:
10070:
10069:
10064:
10062:
10061:
10056:
10050:
10045:
10037:
10023:
10021:
10019:
10018:
10013:
10008:
10003:
9995:
9977:
9975:
9973:
9972:
9967:
9962:
9961:
9956:
9941:
9939:
9937:
9936:
9931:
9929:
9928:
9923:
9908:
9906:
9904:
9903:
9898:
9896:
9895:
9877:
9875:
9873:
9872:
9867:
9865:
9864:
9843:
9841:
9839:
9838:
9833:
9831:
9830:
9825:
9819:
9814:
9806:
9792:
9790:
9788:
9787:
9782:
9780:
9779:
9757:
9755:
9753:
9752:
9747:
9745:
9731:
9729:
9727:
9726:
9721:
9719:
9718:
9700:
9698:
9696:
9695:
9690:
9688:
9674:
9672:
9670:
9669:
9664:
9662:
9648:
9646:
9644:
9643:
9638:
9636:
9635:
9630:
9624:
9619:
9611:
9585:spectral theorem
9578:
9576:
9574:
9573:
9568:
9566:
9549:
9547:
9545:
9544:
9539:
9534:
9533:
9528:
9522:
9508:
9506:
9504:
9503:
9498:
9496:
9491:
9490:
9485:
9466:
9464:
9462:
9461:
9456:
9454:
9438:
9436:
9434:
9433:
9428:
9423:
9422:
9417:
9411:
9397:
9395:
9393:
9392:
9387:
9385:
9369:
9367:
9365:
9364:
9359:
9354:
9349:
9348:
9343:
9324:
9322:
9320:
9319:
9314:
9312:
9292:
9290:
9289:
9284:
9282:
9275:
9274:
9269:
9260:
9259:
9254:
9248:
9240:
9232:
9231:
9226:
9220:
9219:
9214:
9208:
9202:
9201:
9196:
9190:
9185:
9173:
9172:
9167:
9161:
9149:
9148:
9143:
9134:
9129:
9128:
9123:
9114:
9106:
9105:
9100:
9094:
9089:
9083:
9082:
9077:
9071:
9070:
9065:
9059:
9047:
9042:
9041:
9036:
9017:
9015:
9013:
9012:
9007:
9002:
9001:
8996:
8990:
8985:
8977:
8963:
8961:
8959:
8958:
8953:
8951:
8927:matrix, whereas
8926:
8924:
8922:
8921:
8916:
8889:
8887:
8885:
8884:
8879:
8874:
8860:
8858:
8856:
8855:
8850:
8848:
8834:
8832:
8830:
8829:
8824:
8822:
8808:
8806:
8804:
8803:
8798:
8796:
8782:
8780:
8778:
8777:
8772:
8770:
8756:
8754:
8752:
8751:
8746:
8744:
8730:
8728:
8726:
8725:
8720:
8718:
8704:
8702:
8700:
8699:
8694:
8692:
8678:
8676:
8674:
8673:
8668:
8666:
8665:
8641:
8639:
8637:
8636:
8631:
8629:
8615:
8613:
8611:
8610:
8605:
8603:
8589:
8587:
8585:
8584:
8579:
8565:
8563:
8561:
8560:
8555:
8535:
8533:
8531:
8530:
8525:
8523:
8509:
8507:
8505:
8504:
8499:
8476:
8474:
8472:
8471:
8466:
8446:
8444:
8442:
8441:
8436:
8434:
8420:
8418:
8416:
8415:
8410:
8390:
8388:
8386:
8385:
8380:
8353:
8351:
8349:
8348:
8343:
8338:
8314:
8312:
8310:
8309:
8304:
8287:
8285:
8283:
8282:
8277:
8275:
8261:
8259:
8257:
8256:
8251:
8249:
8235:
8233:
8231:
8230:
8225:
8223:
8208:
8206:
8204:
8203:
8198:
8196:
8195:
8190:
8175:
8173:
8171:
8170:
8165:
8163:
8162:
8157:
8135:
8133:
8131:
8130:
8125:
8120:
8106:
8104:
8102:
8101:
8096:
8094:
8093:
8075:
8073:
8071:
8070:
8065:
8063:
8047:
8045:
8043:
8042:
8037:
8032:
8018:
8016:
8014:
8013:
8008:
8006:
8005:
7987:
7985:
7983:
7982:
7977:
7975:
7955:
7953:
7951:
7950:
7945:
7931:
7929:
7927:
7926:
7921:
7919:
7905:
7903:
7901:
7900:
7895:
7871:
7869:
7867:
7866:
7861:
7859:
7845:
7843:
7841:
7840:
7835:
7833:
7819:
7817:
7815:
7814:
7809:
7774:
7772:
7770:
7769:
7764:
7759:
7745:
7743:
7741:
7740:
7735:
7733:
7717:
7715:
7714:
7709:
7707:
7706:
7701:
7695:
7690:
7682:
7665:
7663:
7661:
7660:
7655:
7630:
7628:
7626:
7625:
7620:
7618:
7604:
7602:
7600:
7599:
7594:
7592:
7576:
7574:
7573:
7568:
7566:
7559:
7544:
7539:
7538:
7533:
7520:
7505:
7484:
7482:
7480:
7479:
7474:
7472:
7471:
7453:
7451:
7449:
7448:
7443:
7441:
7427:
7425:
7423:
7422:
7417:
7415:
7414:
7396:
7394:
7392:
7391:
7386:
7384:
7370:
7368:
7366:
7365:
7360:
7358:
7338:
7336:
7334:
7333:
7328:
7299:
7297:
7296:
7291:
7289:
7288:
7280:
7275:
7269:
7264:
7244:
7239:
7228:
7223:
7214:
7209:
7189:
7184:
7176:
7171:
7144:
7139:
7051:
7050:
7045:
7030:
7028:
7026:
7025:
7020:
7018:
7002:
7000:
6999:
6994:
6992:
6988:
6987:
6982:
6973:
6972:
6819:
6818:
6813:
6807:
6798:
6797:
6792:
6783:
6782:
6676:
6675:
6670:
6664:
6642:
6640:
6638:
6637:
6632:
6630:
6629:
6624:
6609:
6607:
6605:
6604:
6599:
6597:
6575:
6573:
6571:
6570:
6565:
6563:
6562:
6557:
6542:
6540:
6538:
6537:
6532:
6530:
6516:
6514:
6512:
6511:
6506:
6504:
6488:
6486:
6485:
6480:
6478:
6474:
6473:
6465:
6460:
6433:
6428:
6340:
6335:
6305:
6300:
6239:
6238:
6233:
6223:
6222:
6214:
6213:
6203:
6177:
6176:
6144:
6143:
6124:
6119:
6107:
6106:
6064:
6055:
6054:
5904:
5887:
5885:
5883:
5882:
5877:
5875:
5874:
5869:
5863:
5858:
5842:
5840:
5839:
5834:
5832:
5831:
5709:
5695:
5693:
5691:
5690:
5685:
5658:
5656:
5654:
5653:
5648:
5631:
5629:
5627:
5626:
5621:
5619:
5618:
5613:
5604:
5596:
5582:
5580:
5578:
5577:
5572:
5546:
5544:
5542:
5541:
5536:
5534:
5520:
5518:
5516:
5515:
5510:
5484:
5482:
5480:
5479:
5474:
5472:
5471:
5466:
5457:
5443:
5441:
5439:
5438:
5433:
5410:
5408:
5406:
5405:
5400:
5398:
5381:
5379:
5377:
5376:
5371:
5366:
5365:
5360:
5345:
5343:
5341:
5340:
5335:
5333:
5332:
5327:
5312:
5310:
5308:
5307:
5302:
5288:
5286:
5284:
5283:
5278:
5276:
5275:
5270:
5255:
5253:
5251:
5250:
5245:
5222:
5220:
5218:
5217:
5212:
5198:
5196:
5194:
5193:
5188:
5161:
5159:
5157:
5156:
5151:
5146:
5145:
5140:
5121:
5119:
5117:
5116:
5111:
5097:
5095:
5093:
5092:
5087:
5082:
5081:
5076:
5061:
5059:
5057:
5056:
5051:
5034:
5032:
5030:
5029:
5024:
5010:
5008:
5006:
5005:
5000:
4995:
4994:
4976:
4974:
4972:
4971:
4966:
4952:
4950:
4948:
4947:
4942:
4940:
4939:
4921:
4919:
4917:
4916:
4911:
4897:
4895:
4893:
4892:
4887:
4873:
4871:
4869:
4868:
4863:
4861:
4860:
4842:
4840:
4838:
4837:
4832:
4830:
4829:
4811:
4809:
4807:
4806:
4801:
4799:
4798:
4786:
4785:
4759:
4757:
4755:
4754:
4749:
4711:
4709:
4707:
4706:
4701:
4690:
4689:
4684:
4663:
4661:
4659:
4658:
4653:
4648:
4634:
4632:
4630:
4629:
4624:
4610:
4608:
4606:
4605:
4600:
4598:
4597:
4577:
4575:
4574:
4569:
4524:
4523:
4518:
4512:
4511:
4496:
4495:
4490:
4467:
4465:
4464:
4459:
4457:
4454:
4447:
4428:
4427:
4411:
4410:
4368:
4366:
4364:
4363:
4358:
4356:
4355:
4337:
4335:
4333:
4332:
4327:
4325:
4311:
4309:
4307:
4306:
4301:
4299:
4298:
4293:
4278:
4277:
4272:
4258:and the columns
4257:
4255:
4253:
4252:
4247:
4245:
4244:
4222:
4220:
4218:
4217:
4212:
4210:
4196:
4194:
4192:
4191:
4186:
4184:
4183:
4178:
4163:
4162:
4157:
4142:
4140:
4138:
4137:
4132:
4130:
4116:
4114:
4112:
4111:
4106:
4104:
4081:
4079:
4077:
4076:
4071:
4069:
4055:
4053:
4051:
4050:
4045:
4043:
4029:
4027:
4025:
4024:
4019:
3996:
3994:
3992:
3991:
3986:
3984:
3966:
3964:
3962:
3961:
3956:
3954:
3953:
3948:
3933:
3931:
3929:
3928:
3923:
3921:
3907:
3905:
3903:
3902:
3897:
3880:
3878:
3876:
3875:
3870:
3868:
3867:
3862:
3843:
3841:
3839:
3838:
3833:
3831:
3817:
3815:
3813:
3812:
3807:
3784:
3782:
3780:
3779:
3774:
3772:
3754:
3752:
3750:
3749:
3744:
3742:
3728:
3726:
3724:
3723:
3718:
3687:
3685:
3683:
3682:
3677:
3675:
3661:
3659:
3657:
3656:
3651:
3646:
3632:
3630:
3628:
3627:
3622:
3620:
3606:
3604:
3602:
3601:
3596:
3594:
3579:Hermitian matrix
3573:
3571:
3569:
3568:
3563:
3561:
3543:
3541:
3539:
3538:
3533:
3531:
3530:
3525:
3510:
3508:
3506:
3505:
3500:
3495:
3482:
3480:
3478:
3477:
3472:
3467:
3466:
3461:
3447:
3445:
3443:
3442:
3437:
3432:
3418:
3416:
3414:
3413:
3408:
3403:
3389:
3387:
3385:
3384:
3379:
3377:
3376:
3371:
3356:
3354:
3352:
3351:
3346:
3341:
3340:
3335:
3329:
3328:
3310:
3308:
3306:
3305:
3300:
3298:
3297:
3292:
3277:
3275:
3273:
3272:
3267:
3265:
3243:
3241:
3239:
3238:
3233:
3231:
3230:
3225:
3210:
3208:
3206:
3205:
3200:
3198:
3180:
3176:
3164:
3162:
3160:
3159:
3154:
3136:
3134:
3132:
3131:
3126:
3112:
3110:
3108:
3107:
3102:
3078:
3076:
3074:
3073:
3068:
3051:
3049:
3047:
3046:
3041:
3017:
3015:
3013:
3012:
3007:
2978:
2976:
2974:
2973:
2968:
2963:
2962:
2957:
2942:
2940:
2938:
2937:
2932:
2930:
2929:
2924:
2909:
2907:
2905:
2904:
2899:
2870:
2868:
2866:
2865:
2860:
2855:
2841:
2839:
2837:
2836:
2831:
2826:
2825:
2820:
2805:
2803:
2801:
2800:
2795:
2793:
2792:
2787:
2772:
2770:
2768:
2767:
2762:
2760:
2759:
2754:
2739:
2737:
2735:
2734:
2729:
2727:
2713:
2711:
2709:
2708:
2703:
2698:
2697:
2692:
2677:
2675:
2673:
2672:
2667:
2665:
2664:
2659:
2644:
2642:
2640:
2639:
2634:
2611:
2609:
2607:
2606:
2601:
2581:
2579:
2577:
2576:
2571:
2569:
2551:
2549:
2547:
2546:
2541:
2539:
2538:
2533:
2518:
2516:
2514:
2513:
2508:
2506:
2492:
2490:
2488:
2487:
2482:
2480:
2464:
2462:
2460:
2458:
2457:
2452:
2450:
2434:
2432:
2430:
2428:
2427:
2422:
2420:
2401:
2399:
2397:
2395:
2394:
2389:
2387:
2386:
2381:
2356:
2354:
2352:
2351:
2346:
2344:
2343:
2338:
2323:
2321:
2319:
2318:
2313:
2308:
2307:
2289:
2287:
2285:
2284:
2279:
2277:
2276:
2271:
2252:
2250:
2248:
2247:
2242:
2240:
2218:
2216:
2214:
2213:
2208:
2206:
2205:
2200:
2185:
2183:
2181:
2180:
2175:
2173:
2159:
2157:
2155:
2154:
2149:
2144:
2143:
2138:
2123:
2121:
2119:
2118:
2113:
2111:
2100:
2083:
2081:
2079:
2078:
2073:
2068:
2050:
2048:
2046:
2045:
2040:
2020:
2018:
2016:
2015:
2010:
2008:
2007:
2002:
1987:
1985:
1983:
1982:
1977:
1975:
1957:
1955:
1953:
1952:
1947:
1927:
1925:
1923:
1922:
1917:
1915:
1884:
1877:
1870:
1856:
1847:
1838:
1832:
1830:
1829:
1824:
1822:
1807:
1798:
1788:
1774:
1719:
1717:
1716:
1711:
1706:
1705:
1700:
1691:
1686:
1685:
1680:
1671:
1666:
1665:
1660:
1644:
1642:
1640:
1639:
1634:
1614:
1612:
1611:
1606:
1604:
1589:
1587:
1585:
1584:
1579:
1559:
1557:
1555:
1554:
1549:
1547:
1533:
1531:
1529:
1528:
1523:
1518:
1504:
1502:
1500:
1499:
1494:
1459:
1457:
1455:
1454:
1449:
1426:
1424:
1422:
1421:
1416:
1414:
1400:
1398:
1396:
1395:
1390:
1388:
1387:
1382:
1367:
1347:
1345:
1343:
1342:
1337:
1332:
1318:
1316:
1314:
1313:
1308:
1306:
1292:
1290:
1288:
1287:
1282:
1280:
1266:
1264:
1263:
1258:
1256:
1244:
1242:
1241:
1236:
1234:
1233:
1212:
1210:
1208:
1207:
1202:
1197:
1183:
1181:
1180:
1175:
1140:
1138:
1137:
1132:
1126:
1121:
1116:
1110:
1109:
1104:
1098:
1097:
1087:
1082:
1064:
1050:
1048:
1047:
1042:
1040:
1039:
1023:
1021:
1019:
1018:
1013:
1008:
1007:
1002:
987:
986:
981:
966:
964:
962:
961:
956:
954:
953:
948:
933:
932:
927:
909:
907:
905:
904:
899:
897:
883:
881:
879:
878:
873:
871:
857:
855:
853:
852:
847:
845:
831:
829:
827:
826:
821:
819:
801:
799:
797:
796:
791:
789:
771:
769:
767:
766:
761:
759:
745:
743:
742:
737:
735:
723:
721:
720:
715:
713:
712:
697:
696:
678:
676:
675:
670:
665:
664:
663:
657:
651:
646:
630:
628:
626:
625:
620:
618:
604:
602:
600:
599:
594:
592:
578:
576:
574:
573:
568:
566:
552:
550:
548:
547:
542:
540:
522:
520:
519:
514:
512:
511:
506:
493:
491:
490:
485:
467:
465:
463:
462:
457:
455:
438:
436:
435:
430:
412:
410:
409:
404:
402:
386:
384:
382:
381:
376:
356:
354:
352:
351:
346:
344:
330:
328:
327:
322:
317:
316:
315:
296:
284:
282:
280:
279:
274:
272:
258:
256:
255:
250:
225:
223:
221:
220:
215:
153:
141:
132:
123:
111:
102:
93:
87:
75:
66:
57:
53:
40:
34:
30:
32433:
32432:
32428:
32427:
32426:
32424:
32423:
32422:
32383:
32382:
32381:
32376:
32337:Spectral method
32322:Ramanujan graph
32270:
32254:
32230:Fredholm theory
32198:
32193:Shilov boundary
32189:Structure space
32167:Generalizations
32162:
32153:Numerical range
32131:
32115:Uniform algebra
32077:
32053:Riesz projector
32038:Min-max theorem
32021:
32007:Direct integral
31963:
31949:Spectral radius
31920:
31875:
31829:
31820:Spectral radius
31768:
31762:Spectral theory
31759:
31729:
31724:
31706:
31670:Advanced topics
31665:
31589:
31568:
31527:
31493:Hilbert–Schmidt
31466:
31457:Gelfand–Naimark
31404:
31354:
31289:
31275:
31245:
31240:
31199:
31195:Multiprocessing
31163:
31159:Sparse problems
31122:
31101:
31096:
31059:
31050:
30994:10.1137/1035134
30959:
30940:
30917:
30892:
30882:Matrix Analysis
30824:
30790:10.1137/0702016
30754:10.1137/0911052
30716:
30643:
30625:
30609:
30604:
30603:
30557:
30553:
30544:
30540:
30517:10.1137/0106005
30501:Hestenes, M. R.
30498:
30494:
30458:
30454:
30449:on 17 June 2019
30446:
30439:
30428:
30424:
30383:(11): 760–766.
30369:
30365:
30312:
30308:
30301:
30273:
30269:
30262:
30258:
30251:
30247:
30240:
30236:
30221:
30217:
30174:
30171:
30170:
30153:
30149:
30143:
30138:
30137:
30132:
30117:
30112:
30111:
30106:
30100:
30095:
30094:
30088:
30083:
30078:
30066:
30063:
30062:
30060:
30056:
30015:
30011:
29972:
29968:
29927:
29923:
29874:
29870:
29858:
29856:
29847:
29846:
29823:
29817:
29813:
29801:
29799:
29790:
29789:
29780:
29766:
29762:
29713:
29709:
29680:
29676:
29647:
29643:
29609:
29603:
29599:
29540:
29536:
29481:
29477:
29422:
29418:
29363:
29359:
29349:
29347:
29337:
29313:
29309:
29282:
29278:
29271:
29267:
29240:J. Neurophysiol
29236:
29232:
29197:Trends Neurosci
29193:
29189:
29184:
29179:
29015:
28970:(or in French,
28968:singular values
28951:
28947:
28945:
28942:
28941:
28881:
28879:
28876:
28875:
28873:
28838:
28814:
28808:
28803:
28802:
28800:
28797:
28796:
28794:
28776:
28774:
28771:
28770:
28768:
28735:
28733:
28730:
28729:
28727:
28705:
28701:
28696:
28691:
28688:
28687:
28685:
28663:
28659:
28654:
28649:
28646:
28645:
28643:
28623:
28621:
28618:
28617:
28615:
28596:
28592:
28590:
28587:
28586:
28584:
28563:
28560:
28559:
28557:
28535:
28531:
28526:
28515:
28511:
28506:
28499:
28495:
28489:
28485:
28479:
28466:
28462:
28457:
28446:
28442:
28437:
28425:
28420:
28419:
28413:
28409:
28404:
28403:
28399:
28393:
28377:
28372:
28371:
28365:
28361:
28356:
28345:
28343:
28340:
28339:
28317:
28314:
28313:
28311:
28293:
28290:
28289:
28287:
28265:
28261:
28256:
28253:
28252:
28250:
28228:
28224:
28219:
28216:
28215:
28213:
28194:
28190:
28188:
28185:
28184:
28182:
28164:
28158:
28153:
28152:
28150:
28147:
28146:
28144:
28126:
28124:
28121:
28120:
28118:
28100:
28097:
28096:
28094:
28076:
28073:
28072:
28070:
28063:
28039:
28034:
28033:
28027:
28023:
28018:
28016:
28013:
28012:
28010:
27991:
27986:
27985:
27980:
27978:
27975:
27974:
27972:
27953:
27948:
27947:
27941:
27937:
27932:
27923:
27918:
27917:
27912:
27904:
27902:
27899:
27898:
27869:
27868:
27862:
27858:
27855:
27854:
27848:
27844:
27837:
27836:
27834:
27831:
27830:
27809:
27805:
27803:
27800:
27799:
27797:
27778:
27774:
27772:
27769:
27768:
27766:
27748:
27746:
27743:
27742:
27740:
27711:
27705:
27700:
27699:
27697:
27694:
27693:
27691:
27672:
27667:
27666:
27660:
27656:
27651:
27649:
27646:
27645:
27643:
27604:
27600:
27598:
27595:
27594:
27592:
27573:
27570:
27569:
27567:
27546:
27542:
27540:
27537:
27536:
27534:
27515:
27510:
27509:
27503:
27499:
27494:
27486:
27484:
27481:
27480:
27460:
27457:
27456:
27454:
27421:
27418:
27417:
27415:
27394:
27392:
27389:
27388:
27386:
27365:
27363:
27360:
27359:
27357:
27333:
27331:
27328:
27327:
27325:
27304:
27301:
27300:
27298:
27280:
27278:
27275:
27274:
27272:
27250:
27245:
27244:
27239:
27234:
27226:
27224:
27221:
27220:
27218:
27215:
27189:
27187:
27184:
27183:
27181:
27160:
27158:
27155:
27154:
27152:
27134:
27132:
27129:
27128:
27126:
27105:
27100:
27095:
27093:
27090:
27089:
27087:
27066:
27064:
27061:
27060:
27058:
27034:
27032:
27029:
27028:
27026:
27005:
27003:
27000:
26999:
26997:
26979:
26977:
26974:
26973:
26971:
26950:
26945:
26940:
26938:
26935:
26934:
26932:
26911:
26909:
26906:
26905:
26903:
26885:
26883:
26880:
26879:
26877:
26874:
26869:
26836:
26831:
26830:
26821:
26817:
26812:
26803:
26790:
26787:
26786:
26784:
26782:
26779:
26778:
26751:
26747:
26736:
26734:
26731:
26730:
26728:
26707:
26705:
26702:
26701:
26699:
26692:Schatten 2-norm
26664:
26662:
26659:
26658:
26656:
26637:
26633:
26631:
26628:
26627:
26625:
26604:
26599:
26589:
26576:
26573:
26572:
26570:
26559:
26554:
26551:
26550:
26520:
26514:
26509:
26508:
26507:
26503:
26495:
26482:
26474:
26469:
26458:
26453:
26450:
26449:
26421:
26415:
26410:
26409:
26408:
26404:
26387:
26379:
26374:
26371:
26370:
26344:
26341:
26340:
26338:
26335:Hilbert–Schmidt
26331:
26308:
26302:
26297:
26296:
26294:
26291:
26290:
26288:
26267:
26263:
26259:
26254:
26248:
26243:
26242:
26222:
26217:
26214:
26213:
26186:
26184:
26181:
26180:
26178:
26152:
26148:
26144:
26139:
26133:
26128:
26127:
26122:
26119:
26118:
26116:
26095:
26091:
26087:
26082:
26076:
26071:
26070:
26065:
26062:
26061:
26034:
26030:
26026:
26021:
26015:
26010:
26009:
26004:
26001:
26000:
25998:
25974:
25970:
25965:
25959:
25954:
25953:
25939:
25934:
25931:
25930:
25910:
25908:
25905:
25904:
25902:
25884:
25880:
25878:
25875:
25874:
25852:
25848:
25846:
25843:
25842:
25840:
25821:
25817:
25815:
25812:
25811:
25809:
25791:
25789:
25786:
25785:
25783:
25756:
25754:
25751:
25750:
25748:
25730:
25727:
25726:
25724:
25699:
25697:
25694:
25693:
25691:
25673:
25670:
25669:
25667:
25666:The sum of the
25664:
25659:
25629:
25627:
25624:
25623:
25621:
25600:
25598:
25595:
25594:
25592:
25558:
25555:
25554:
25552:
25533:
25528:
25527:
25525:
25522:
25521:
25519:
25501:
25498:
25497:
25495:
25476:
25471:
25470:
25468:
25465:
25464:
25462:
25461:row vectors of
25444:
25441:
25440:
25438:
25420:
25418:
25415:
25414:
25412:
25394:
25391:
25390:
25388:
25361:
25358:
25357:
25355:
25336:
25331:
25326:
25323:
25320:
25319:
25317:
25293:
25290:
25289:
25287:
25268:
25263:
25262:
25260:
25257:
25256:
25254:
25228:
25225:
25224:
25222:
25203:
25198:
25197:
25195:
25192:
25191:
25189:
25167:
25162:
25157:
25150:
25145:
25144:
25138:
25133:
25132:
25118:
25116:
25115:
25113:
25110:
25109:
25090:
25087:
25086:
25084:
25060:
25058:
25057:
25055:
25052:
25051:
25049:
25025:
25023:
25020:
25019:
25017:
24993:
24990:
24989:
24987:
24969:
24966:
24965:
24963:
24960:
24929:
24926:
24925:
24923:
24904:
24899:
24894:
24891:
24888:
24887:
24885:
24861:
24858:
24857:
24855:
24836:
24831:
24830:
24828:
24825:
24824:
24822:
24796:
24793:
24792:
24790:
24771:
24766:
24765:
24763:
24760:
24759:
24757:
24715:
24712:
24711:
24709:
24690:
24685:
24684:
24682:
24679:
24678:
24676:
24658:
24656:
24653:
24652:
24650:
24631:
24626:
24625:
24623:
24620:
24619:
24617:
24598:
24593:
24592:
24590:
24587:
24586:
24584:
24583:row vectors of
24566:
24563:
24562:
24560:
24542:
24540:
24537:
24536:
24534:
24516:
24513:
24512:
24510:
24488:
24483:
24478:
24471:
24466:
24465:
24459:
24454:
24453:
24445:
24443:
24440:
24439:
24419:
24417:
24414:
24413:
24411:
24408:
24382:
24380:
24377:
24376:
24374:
24328:
24325:
24324:
24322:
24293:
24290:
24289:
24287:
24268:
24263:
24258:
24255:
24252:
24251:
24249:
24225:
24222:
24221:
24219:
24200:
24195:
24194:
24192:
24189:
24188:
24186:
24160:
24157:
24156:
24154:
24135:
24130:
24129:
24127:
24124:
24123:
24121:
24100:
24098:
24095:
24094:
24092:
24074:
24071:
24070:
24068:
24049:
24044:
24043:
24041:
24038:
24037:
24035:
24014:
24012:
24009:
24008:
24006:
23988:
23986:
23983:
23982:
23980:
23962:
23959:
23958:
23956:
23937:
23932:
23931:
23929:
23926:
23925:
23923:
23904:
23899:
23898:
23896:
23893:
23892:
23890:
23850:
23847:
23846:
23824:
23819:
23814:
23807:
23802:
23801:
23795:
23790:
23789:
23781:
23779:
23776:
23775:
23755:
23753:
23750:
23749:
23747:
23744:
23721:
23718:
23717:
23715:
23697:
23695:
23692:
23691:
23689:
23665:
23662:
23661:
23659:
23648:
23642:
23636:
23630:
23624:
23618:
23611:
23593:
23592:
23582:
23578:
23572:
23567:
23557:
23553:
23535:
23531:
23525:
23520:
23510:
23506:
23488:
23484:
23478:
23473:
23463:
23459:
23441:
23437:
23431:
23426:
23416:
23412:
23394:
23390:
23384:
23379:
23369:
23365:
23347:
23343:
23337:
23332:
23322:
23318:
23307:
23295:
23290:
23289:
23283:
23279:
23274:
23265:
23260:
23259:
23253:
23249:
23244:
23235:
23230:
23229:
23223:
23219:
23214:
23205:
23200:
23199:
23193:
23189:
23184:
23182:
23173:
23172:
23162:
23157:
23156:
23150:
23145:
23132:
23127:
23114:
23109:
23096:
23091:
23082:
23073:
23067:
23066:
23065:
23059:
23054:
23053:
23047:
23043:
23038:
23029:
23024:
23023:
23017:
23013:
23008:
22999:
22994:
22993:
22987:
22983:
22978:
22969:
22964:
22963:
22957:
22953:
22948:
22942:
22941:
22939:
22930:
22925:
22924:
22918:
22914:
22909:
22900:
22895:
22894:
22888:
22884:
22879:
22870:
22865:
22864:
22858:
22854:
22849:
22840:
22835:
22834:
22828:
22824:
22819:
22817:
22810:
22804:
22800:
22796:
22794:
22791:
22790:
22767:
22763:
22761:
22758:
22757:
22739:
22737:
22734:
22733:
22731:
22715:
22706:
22702:
22700:
22697:
22696:
22677:
22673:
22667:
22663:
22654:
22650:
22644:
22640:
22631:
22627:
22621:
22617:
22609:
22603:
22599:
22591:
22589:
22586:
22585:
22561:
22558:
22557:
22555:
22552:
22520:
22512:
22510:
22507:
22506:
22504:
22482:
22477:
22476:
22471:
22466:
22458:
22456:
22453:
22452:
22450:
22428:
22423:
22422:
22417:
22409:
22407:
22404:
22403:
22401:
22383:
22381:
22378:
22377:
22375:
22353:
22348:
22340:
22338:
22335:
22334:
22332:
22289:
22288:
22283:
22281:
22275:
22270:
22269:
22266:
22265:
22260:
22258:
22253:
22246:
22245:
22243:
22240:
22239:
22216:
22210:
22205:
22204:
22202:
22199:
22198:
22196:
22175:
22170:
22169:
22164:
22162:
22159:
22158:
22156:
22138:
22136:
22133:
22132:
22130:
22126:, Lecture 31).
22091:
22088:
22087:
22085:
22057:
22054:
22053:
22051:
22016:, Lecture 31).
21994:
21990:
21978:
21974:
21966:
21963:
21962:
21960:
21938:
21936:
21933:
21932:
21930:
21912:
21909:
21908:
21906:
21888:
21885:
21884:
21882:
21861:
21855:
21851:
21839:
21835:
21827:
21824:
21823:
21821:
21820:for a cost of
21813:, Lecture 31).
21788:
21784:
21773:
21770:
21769:
21767:
21740:
21737:
21736:
21734:
21707:
21704:
21703:
21701:
21698:machine epsilon
21662:
21659:
21658:
21656:
21634:
21630:
21619:
21616:
21615:
21613:
21588:
21586:
21583:
21582:
21580:
21557:
21552:
21551:
21546:
21544:
21541:
21540:
21538:
21520:
21514:
21509:
21508:
21506:
21503:
21502:
21500:
21480:
21478:
21475:
21474:
21456:
21454:
21451:
21450:
21448:
21427:
21421:
21416:
21415:
21413:
21410:
21409:
21407:
21389:
21387:
21384:
21383:
21381:
21359:
21354:
21353:
21348:
21346:
21343:
21342:
21340:
21315:
21313:
21310:
21309:
21307:
21300:
21270:
21267:
21266:
21246:
21242:
21231:
21228:
21227:
21211:
21208:
21207:
21191:
21188:
21187:
21168:
21165:
21164:
21148:
21145:
21144:
21116:
21112:
21104:
21101:
21100:
21091:Givens rotation
21074:
21071:
21070:
21059:
21038:
21035:
21034:
21018:
21015:
21014:
20994:
20991:
20990:
20974:
20971:
20970:
20953:
20949:
20935:
20932:
20931:
20912:
20909:
20908:
20850:
20847:
20846:
20818:
20815:
20814:
20798:
20795:
20794:
20778:
20775:
20774:
20737:
20734:
20733:
20717:
20714:
20713:
20662:
20659:
20658:
20653:Jacobi rotation
20649:
20644:
20614:
20609:
20608:
20606:
20603:
20602:
20600:
20581:
20576:
20575:
20573:
20570:
20569:
20567:
20549:
20547:
20544:
20543:
20541:
20523:
20521:
20518:
20517:
20515:
20494:
20486:
20478:
20475:
20474:
20472:
20453:
20452:
20443:
20438:
20437:
20431:
20427:
20420:
20414:
20409:
20408:
20402:
20397:
20396:
20393:
20392:
20383:
20378:
20377:
20371:
20367:
20360:
20354:
20349:
20348:
20343:
20339:
20337:
20334:
20333:
20309:
20305:
20293:
20289:
20277:
20273:
20271:
20268:
20267:
20240:
20226:
20221:
20218:
20217:
20197:
20196:
20191:
20186:
20183:
20180:
20179:
20177:
20157:
20156:
20151:
20146:
20143:
20140:
20139:
20137:
20121:
20120:
20111:
20106:
20105:
20099:
20095:
20079:
20073:
20068:
20067:
20061:
20056:
20055:
20052:
20051:
20036:
20031:
20030:
20024:
20020:
20010:
20004:
19999:
19998:
19993:
19989:
19987:
19984:
19983:
19963:
19957:
19952:
19951:
19939:
19935:
19927:
19921:
19916:
19915:
19903:
19899:
19891:
19886:
19880:
19875:
19874:
19860:
19857:
19856:
19827:
19823:
19821:
19818:
19817:
19815:
19797:
19795:
19792:
19791:
19789:
19770:
19765:
19764:
19762:
19759:
19758:
19756:
19737:
19732:
19731:
19729:
19726:
19725:
19723:
19699:
19694:
19693:
19691:
19688:
19687:
19685:
19666:
19661:
19660:
19658:
19655:
19654:
19652:
19631:
19623:
19615:
19612:
19611:
19609:
19590:
19586:
19584:
19581:
19580:
19578:
19556:
19551:
19550:
19548:
19545:
19544:
19542:
19523:
19518:
19517:
19515:
19512:
19511:
19509:
19490:
19486:
19484:
19481:
19480:
19478:
19450:
19446:
19444:
19441:
19440:
19438:
19420:
19418:
19415:
19414:
19412:
19387:
19383:
19381:
19378:
19377:
19375:
19357:
19355:
19352:
19351:
19349:
19331:
19328:
19327:
19325:
19292:
19288:
19286:
19283:
19282:
19280:
19255:
19251:
19249:
19246:
19245:
19243:
19215:
19211:
19196:
19192:
19190:
19187:
19186:
19184:
19166:
19163:
19162:
19160:
19132:
19128:
19120:
19118:
19115:
19114:
19089:
19085:
19077:
19075:
19072:
19071:
19053:
19048:
19042:
19037:
19036:
19025:
19017:
19009:
19006:
19005:
18988:
18983:
18982:
18980:
18977:
18976:
18948:
18945:
18944:
18919:
18915:
18913:
18910:
18909:
18907:
18883:
18880:
18879:
18877:
18859:
18857:
18854:
18853:
18851:
18830:
18828:
18825:
18824:
18822:
18804:
18802:
18799:
18798:
18796:
18775:
18773:
18770:
18769:
18767:
18749:
18747:
18744:
18743:
18741:
18720:
18715:
18708:
18707:
18702:
18701:
18699:
18696:
18695:
18693:
18690:
18667:
18661:
18656:
18655:
18653:
18650:
18649:
18647:
18628:
18623:
18622:
18617:
18615:
18612:
18611:
18609:
18591:
18585:
18580:
18579:
18577:
18574:
18573:
18571:
18552:
18547:
18546:
18541:
18539:
18536:
18535:
18533:
18511:
18506:
18505:
18500:
18495:
18487:
18485:
18482:
18481:
18458:
18449:
18444:
18439:
18427:
18422:
18421:
18415:
18410:
18409:
18399:
18398:
18392:
18391:
18385:
18380:
18375:
18363:
18358:
18357:
18350:
18349:
18342:
18341:
18335:
18330:
18329:
18327:
18321:
18316:
18315:
18308:
18307:
18298:
18291:
18290:
18284:
18279:
18278:
18276:
18270:
18265:
18264:
18257:
18256:
18255:
18248:
18247:
18241:
18240:
18233:
18232:
18227:
18221:
18220:
18215:
18204:
18200:
18198:
18191:
18190:
18183:
18182:
18175:
18174:
18168:
18163:
18162:
18160:
18154:
18149:
18148:
18141:
18140:
18138:
18135:
18134:
18110:
18107:
18106:
18090:
18088:
18085:
18084:
18062:
18057:
18056:
18054:
18051:
18050:
18048:
18021:
18020:
18014:
18013:
18006:
18005:
18000:
17994:
17993:
17988:
17977:
17972:
17971:
17964:
17963:
17956:
17955:
17947:
17945:
17942:
17941:
17920:
17915:
17914:
17912:
17909:
17908:
17906:
17887:
17882:
17881:
17879:
17876:
17875:
17873:
17852:
17851:
17845:
17840:
17839:
17837:
17831:
17826:
17825:
17818:
17817:
17809:
17807:
17804:
17803:
17786:
17781:
17780:
17778:
17775:
17774:
17757:
17752:
17751:
17749:
17746:
17745:
17744:the columns in
17722:
17718:
17717:
17702:
17698:
17693:
17692:
17687:
17675:
17671:
17666:
17665:
17650:
17646:
17641:
17640:
17634:
17629:
17628:
17623:
17617:
17612:
17611:
17605:
17600:
17595:
17582:
17578:
17573:
17572:
17563:
17558:
17557:
17551:
17546:
17541:
17538:
17535:
17534:
17516:
17513:
17512:
17496:
17493:
17492:
17476:
17474:
17471:
17470:
17453:
17448:
17447:
17445:
17442:
17441:
17424:
17419:
17418:
17416:
17413:
17412:
17395:
17390:
17389:
17387:
17384:
17383:
17363:
17352:
17343:
17340:
17339:
17332:
17331:
17325:
17320:
17319:
17314:
17304:
17297:
17292:
17282:
17281:
17279:
17276:
17275:
17258:
17253:
17252:
17250:
17247:
17246:
17230:
17224:
17219:
17218:
17216:
17213:
17212:
17195:
17190:
17189:
17187:
17184:
17183:
17166:
17155:
17145:
17140:
17139:
17134:
17131:
17130:
17113:
17108:
17107:
17105:
17102:
17101:
17080:
17069:
17060:
17057:
17056:
17049:
17048:
17042:
17037:
17036:
17031:
17021:
17014:
17009:
16999:
16998:
16996:
16993:
16992:
16975:
16964:
16954:
16949:
16948:
16943:
16938:
16935:
16934:
16917:
16906:
16896:
16892:
16887:
16884:
16883:
16867:
16861:
16856:
16855:
16853:
16850:
16849:
16832:
16821:
16811:
16806:
16805:
16800:
16797:
16796:
16780:
16771:
16766:
16761:
16754:
16749:
16748:
16743:
16741:
16738:
16737:
16718:
16709:
16704:
16703:
16698:
16696:
16693:
16692:
16671:
16662:
16657:
16652:
16642:
16637:
16636:
16631:
16620:
16608:
16603:
16598:
16591:
16586:
16585:
16577:
16569:
16560:
16555:
16550:
16538:
16533:
16532:
16520:
16516:
16511:
16510:
16504:
16499:
16498:
16493:
16484:
16479:
16474:
16462:
16457:
16456:
16450:
16445:
16444:
16442:
16439:
16438:
16408:
16404:
16399:
16398:
16392:
16387:
16386:
16381:
16372:
16367:
16366:
16364:
16361:
16360:
16339:
16338:
16329:
16324:
16323:
16316:
16310:
16305:
16300:
16293:
16288:
16287:
16278:
16273:
16268:
16261:
16256:
16255:
16252:
16251:
16242:
16237:
16236:
16229:
16223:
16218:
16217:
16211:
16206:
16201:
16197:
16196:
16187:
16182:
16181:
16174:
16168:
16163:
16162:
16156:
16151:
16146:
16141:
16139:
16136:
16135:
16116:
16111:
16110:
16108:
16105:
16104:
16087:
16082:
16081:
16079:
16076:
16075:
16059:
16057:
16054:
16053:
16037:
16028:
16023:
16022:
16017:
16015:
16012:
16011:
15990:
15981:
15976:
15975:
15970:
15964:
15959:
15958:
15952:
15947:
15942:
15932:
15923:
15918:
15917:
15912:
15906:
15901:
15900:
15894:
15889:
15884:
15881:
15878:
15877:
15852:
15851:
15846:
15840:
15839:
15834:
15829:
15822:
15821:
15811:
15810:
15804:
15799:
15798:
15793:
15787:
15782:
15781:
15775:
15770:
15765:
15762:
15756:
15751:
15750:
15745:
15739:
15734:
15733:
15727:
15722:
15717:
15713:
15712:
15706:
15701:
15700:
15695:
15689:
15684:
15683:
15677:
15672:
15667:
15664:
15658:
15653:
15652:
15647:
15641:
15636:
15635:
15629:
15624:
15619:
15611:
15610:
15600:
15599:
15593:
15588:
15587:
15583:
15577:
15572:
15571:
15564:
15563:
15557:
15551:
15546:
15545:
15538:
15537:
15531:
15526:
15521:
15517:
15516:
15510:
15505:
15500:
15492:
15491:
15489:
15486:
15485:
15467:
15465:
15462:
15461:
15445:
15439:
15434:
15433:
15431:
15428:
15427:
15410:
15405:
15404:
15402:
15399:
15398:
15381:
15376:
15375:
15373:
15370:
15369:
15351:
15350:
15344:
15339:
15338:
15336:
15330:
15325:
15324:
15317:
15316:
15308:
15306:
15303:
15302:
15286:
15284:
15281:
15280:
15254:
15243:
15241:
15240:
15239:
15237:
15234:
15233:
15217:
15211:
15206:
15205:
15203:
15200:
15199:
15177:
15174:
15173:
15157:
15155:
15152:
15151:
15135:
15132:
15131:
15111:
15100:
15098:
15097:
15096:
15094:
15091:
15090:
15074:
15068:
15063:
15062:
15060:
15057:
15056:
15040:
15037:
15036:
15020:
15017:
15016:
15000:
14998:
14995:
14994:
14957:
14954:
14953:
14937:
14931:
14926:
14925:
14923:
14920:
14919:
14903:
14900:
14899:
14877:
14874:
14873:
14857:
14855:
14852:
14851:
14828:
14827:
14822:
14816:
14815:
14810:
14805:
14798:
14797:
14783:
14781:
14780:
14772:
14767:
14761:
14756:
14755:
14749:
14744:
14743:
14741:
14738:
14737:
14719:
14717:
14714:
14713:
14712:unitary matrix
14689:
14686:
14685:
14683:
14663:
14657:
14652:
14651:
14649:
14646:
14645:
14621:
14618:
14617:
14615:
14599:
14597:
14594:
14593:
14590:
14564:
14562:
14559:
14558:
14556:
14532:
14529:
14528:
14526:
14508:
14503:
14497:
14492:
14491:
14480:
14472:
14469:
14468:
14466:
14448:
14446:
14443:
14442:
14440:
14419:
14417:
14414:
14413:
14411:
14393:
14390:
14389:
14387:
14363:
14355:
14352:
14351:
14349:
14331:
14329:
14326:
14325:
14323:
14305:
14303:
14300:
14299:
14297:
14276:
14274:
14271:
14270:
14268:
14250:
14248:
14245:
14244:
14242:
14221:
14210:
14205:
14203:
14200:
14199:
14197:
14176:
14168:
14157:
14143:
14137:
14132:
14131:
14114:
14109:
14103:
14098:
14097:
14092:
14089:
14088:
14068:
14066:
14063:
14062:
14060:
14040:
14038:
14035:
14034:
14032:
14031:
14009:
14006:
14005:
14003:
13982:
13979:
13978:
13976:
13952:
13946:
13941:
13940:
13923:
13918:
13912:
13907:
13906:
13901:
13898:
13897:
13877:
13875:
13872:
13871:
13869:
13834:
13826:
13823:
13822:
13804:
13802:
13799:
13798:
13796:
13773:
13772:
13767:
13762:
13756:
13751:
13750:
13743:
13738:
13735:
13734:
13729:
13722:
13716:
13711:
13710:
13706:
13702:
13694:
13691:
13690:
13661:
13658:
13657:
13655:
13637:
13635:
13632:
13631:
13629:
13607:
13605:
13602:
13601:
13599:
13578:
13567:
13562:
13560:
13557:
13556:
13554:
13536:
13534:
13531:
13530:
13528:
13510:
13507:
13506:
13504:
13501:
13444:Lanczos methods
13420:
13418:
13415:
13414:
13382:
13378:
13373:
13367:
13363:
13355:
13352:
13351:
13308:
13288:
13279:Wahba's problem
13271:
13250:
13244:
13239:
13238:
13230:
13228:
13225:
13224:
13204:
13200:
13192:
13189:
13188:
13167:
13159:
13153:
13148:
13147:
13141:
13134:
13130:
13125:
13117:
13112:
13099:
13091:
13089:
13086:
13085:
13062:
13060:
13057:
13056:
13054:
13036:
13034:
13031:
13030:
13028:
13010:
13008:
13005:
13004:
13002:
12975:
12966:
12965:
12957:
12952:
12944:
12942:
12939:
12938:
12919:
12914:
12913:
12908:
12900:
12898:
12895:
12894:
12892:
12873:
12868:
12867:
12862:
12854:
12852:
12849:
12848:
12846:
12827:
12822:
12821:
12816:
12811:
12803:
12801:
12798:
12797:
12795:
12777:
12775:
12772:
12771:
12769:
12750:
12745:
12744:
12739:
12734:
12732:
12729:
12728:
12726:
12704:
12699:
12698:
12693:
12691:
12688:
12687:
12685:
12664:
12656:
12654:
12651:
12650:
12648:
12623:
12621:
12618:
12617:
12615:
12597:
12595:
12592:
12591:
12589:
12568:
12566:
12563:
12562:
12560:
12557:
12527:
12522:
12512:
12507:
12501:
12496:
12490:
12482:
12479:
12478:
12458:
12456:
12453:
12452:
12450:
12432:
12430:
12427:
12426:
12424:
12396:
12392:
12390:
12387:
12386:
12384:
12365:
12360:
12359:
12357:
12354:
12353:
12351:
12332:
12328:
12326:
12323:
12322:
12320:
12302:
12299:
12298:
12296:
12277:
12272:
12271:
12269:
12266:
12265:
12263:
12244:
12239:
12238:
12236:
12233:
12232:
12230:
12208:
12203:
12202:
12193:
12188:
12187:
12181:
12177:
12171:
12158:
12153:
12152:
12146:
12134:
12132:
12129:
12128:
12108:
12106:
12103:
12102:
12100:
12078:
12074:
12068:
12064:
12052:
12048:
12046:
12043:
12042:
12040:
12019:
12011:
12003:
12001:
11998:
11997:
11995:
11994:of two vectors
11973:
11971:
11968:
11967:
11965:
11962:
11929:
11926:
11925:
11923:
11907:
11905:
11902:
11901:
11879:
11877:
11876:
11874:
11871:
11870:
11848:
11843:
11842:
11831:
11829:
11828:
11823:
11809:
11807:
11806:
11804:
11801:
11800:
11777:
11775:
11772:
11771:
11769:
11743:
11742:
11731:
11729:
11728:
11722:
11721:
11713:
11710:
11709:
11685:
11683:
11682:
11680:
11677:
11676:
11674:
11656:
11654:
11651:
11650:
11648:
11626:
11623:
11622:
11620:
11594:
11592:
11591:
11589:
11586:
11585:
11567:
11565:
11562:
11561:
11559:
11552:
11523:
11521:
11518:
11517:
11499:
11497:
11494:
11493:
11491:
11464:
11462:
11459:
11458:
11456:
11437:
11432:
11431:
11429:
11426:
11425:
11423:
11398:
11396:
11393:
11392:
11390:
11372:
11370:
11367:
11366:
11364:
11346:
11344:
11341:
11340:
11338:
11320:
11318:
11315:
11314:
11312:
11291:
11289:
11286:
11285:
11283:
11272:
11249:
11247:
11244:
11243:
11241:
11216:
11211:
11208:
11207:
11189:
11184:
11182:
11179:
11178:
11176:
11154:
11152:
11149:
11148:
11146:
11139:
11114:
11112:
11109:
11108:
11106:
11085:
11083:
11080:
11079:
11077:
11058:
11053:
11052:
11050:
11047:
11046:
11044:
11026:
11018:
11012:
11007:
11006:
11004:
11001:
11000:
10998:
10980:
10978:
10975:
10974:
10972:
10953:
10951:
10948:
10947:
10945:
10923:
10921:
10918:
10917:
10915:
10897:
10895:
10892:
10891:
10889:
10871:
10869:
10866:
10865:
10863:
10842:
10840:
10837:
10836:
10834:
10812:
10810:
10807:
10806:
10804:
10786:
10784:
10781:
10780:
10778:
10760:
10758:
10755:
10754:
10752:
10734:
10732:
10729:
10728:
10726:
10705:
10703:
10700:
10699:
10697:
10679:
10677:
10674:
10673:
10671:
10653:
10645:
10640:
10638:
10635:
10634:
10632:
10625:
10596:
10594:
10591:
10590:
10573:
10568:
10567:
10565:
10562:
10561:
10539:
10534:
10533:
10527:
10522:
10521:
10516:
10507:
10502:
10501:
10499:
10496:
10495:
10474:
10469:
10468:
10463:
10458:
10450:
10448:
10445:
10444:
10442:
10424:
10422:
10419:
10418:
10416:
10409:
10404:
10375:
10372:
10371:
10369:
10344:
10342:
10339:
10338:
10336:
10318:
10316:
10313:
10312:
10310:
10292:
10290:
10287:
10286:
10284:
10266:
10264:
10261:
10260:
10258:
10240:
10238:
10235:
10234:
10232:
10214:
10212:
10209:
10208:
10206:
10188:
10186:
10183:
10182:
10180:
10162:
10160:
10157:
10156:
10154:
10133:
10131:
10128:
10127:
10125:
10103:
10098:
10097:
10092:
10084:
10082:
10079:
10078:
10076:
10057:
10052:
10051:
10046:
10041:
10033:
10031:
10028:
10027:
10025:
10004:
9999:
9991:
9989:
9986:
9985:
9983:
9957:
9952:
9951:
9949:
9946:
9945:
9943:
9924:
9919:
9918:
9916:
9913:
9912:
9910:
9891:
9887:
9885:
9882:
9881:
9879:
9857:
9853:
9851:
9848:
9847:
9845:
9826:
9821:
9820:
9815:
9810:
9802:
9800:
9797:
9796:
9794:
9775:
9771:
9769:
9766:
9765:
9763:
9741:
9739:
9736:
9735:
9733:
9714:
9710:
9708:
9705:
9704:
9702:
9684:
9682:
9679:
9678:
9676:
9658:
9656:
9653:
9652:
9650:
9631:
9626:
9625:
9620:
9615:
9607:
9605:
9602:
9601:
9599:
9562:
9560:
9557:
9556:
9554:
9529:
9524:
9523:
9518:
9516:
9513:
9512:
9510:
9492:
9486:
9481:
9480:
9478:
9475:
9474:
9472:
9450:
9448:
9445:
9444:
9442:
9418:
9413:
9412:
9407:
9405:
9402:
9401:
9399:
9381:
9379:
9376:
9375:
9373:
9372:The columns of
9350:
9344:
9339:
9338:
9336:
9333:
9332:
9330:
9308:
9306:
9303:
9302:
9300:
9299:The columns of
9280:
9279:
9270:
9265:
9264:
9255:
9250:
9249:
9244:
9236:
9227:
9222:
9221:
9215:
9210:
9209:
9204:
9197:
9192:
9191:
9186:
9181:
9174:
9168:
9163:
9162:
9157:
9154:
9153:
9144:
9139:
9138:
9130:
9124:
9119:
9118:
9110:
9101:
9096:
9095:
9090:
9085:
9078:
9073:
9072:
9066:
9061:
9060:
9055:
9048:
9043:
9037:
9032:
9031:
9027:
9025:
9022:
9021:
8997:
8992:
8991:
8986:
8981:
8973:
8971:
8968:
8967:
8965:
8947:
8945:
8942:
8941:
8939:
8904:
8901:
8900:
8898:
8895:
8870:
8868:
8865:
8864:
8862:
8844:
8842:
8839:
8838:
8836:
8818:
8816:
8813:
8812:
8810:
8792:
8790:
8787:
8786:
8784:
8766:
8764:
8761:
8760:
8758:
8740:
8738:
8735:
8734:
8732:
8714:
8712:
8709:
8708:
8706:
8688:
8686:
8683:
8682:
8680:
8658:
8654:
8652:
8649:
8648:
8646:
8625:
8623:
8620:
8619:
8617:
8599:
8597:
8594:
8593:
8591:
8573:
8570:
8569:
8567:
8543:
8540:
8539:
8537:
8519:
8517:
8514:
8513:
8511:
8484:
8481:
8480:
8478:
8454:
8451:
8450:
8448:
8447:is padded with
8430:
8428:
8425:
8424:
8422:
8398:
8395:
8394:
8392:
8365:
8362:
8361:
8359:
8334:
8332:
8329:
8328:
8326:
8295:
8292:
8291:
8289:
8271:
8269:
8266:
8265:
8263:
8245:
8243:
8240:
8239:
8237:
8219:
8217:
8214:
8213:
8211:
8191:
8186:
8185:
8183:
8180:
8179:
8177:
8158:
8153:
8152:
8150:
8147:
8146:
8144:
8116:
8114:
8111:
8110:
8108:
8089:
8085:
8083:
8080:
8079:
8077:
8059:
8057:
8054:
8053:
8051:
8028:
8026:
8023:
8022:
8020:
8001:
7997:
7995:
7992:
7991:
7989:
7971:
7969:
7966:
7965:
7963:
7939:
7936:
7935:
7933:
7915:
7913:
7910:
7909:
7907:
7883:
7880:
7879:
7877:
7855:
7853:
7850:
7849:
7847:
7829:
7827:
7824:
7823:
7821:
7782:
7779:
7778:
7776:
7755:
7753:
7750:
7749:
7747:
7729:
7727:
7724:
7723:
7721:
7702:
7697:
7696:
7691:
7686:
7678:
7676:
7673:
7672:
7646:
7643:
7642:
7640:
7614:
7612:
7609:
7608:
7606:
7588:
7586:
7583:
7582:
7580:
7564:
7563:
7555:
7545:
7540:
7534:
7529:
7528:
7525:
7524:
7516:
7506:
7498:
7494:
7492:
7489:
7488:
7467:
7463:
7461:
7458:
7457:
7455:
7437:
7435:
7432:
7431:
7429:
7410:
7406:
7404:
7401:
7400:
7398:
7380:
7378:
7375:
7374:
7372:
7354:
7352:
7349:
7348:
7346:
7322:
7319:
7318:
7316:
7313:
7308:
7283:
7282:
7274:
7271:
7263:
7260:
7253:
7246:
7238:
7231:
7230:
7222:
7216:
7208:
7205:
7198:
7191:
7183:
7179:
7178:
7170:
7167:
7160:
7153:
7146:
7138:
7134:
7133:
7126:
7119:
7112:
7105:
7097:
7096:
7089:
7082:
7075:
7068:
7056:
7055:
7046:
7041:
7040:
7038:
7035:
7034:
7014:
7012:
7009:
7008:
7006:
6990:
6989:
6983:
6978:
6977:
6967:
6966:
6961:
6956:
6951:
6946:
6940:
6939:
6934:
6929:
6924:
6919:
6913:
6912:
6907:
6902:
6897:
6892:
6886:
6885:
6880:
6875:
6870:
6865:
6859:
6858:
6853:
6848:
6843:
6838:
6828:
6827:
6820:
6814:
6809:
6808:
6803:
6800:
6799:
6793:
6788:
6787:
6777:
6776:
6771:
6766:
6761:
6755:
6754:
6749:
6744:
6739:
6733:
6732:
6727:
6722:
6717:
6711:
6710:
6705:
6700:
6695:
6685:
6684:
6677:
6671:
6666:
6665:
6660:
6656:
6654:
6651:
6650:
6625:
6620:
6619:
6617:
6614:
6613:
6611:
6593:
6591:
6588:
6587:
6585:
6558:
6553:
6552:
6550:
6547:
6546:
6544:
6526:
6524:
6521:
6520:
6518:
6500:
6498:
6495:
6494:
6492:
6476:
6475:
6468:
6467:
6459:
6456:
6449:
6442:
6435:
6427:
6420:
6419:
6412:
6405:
6398:
6391:
6383:
6382:
6375:
6368:
6361:
6351:
6343:
6342:
6334:
6328:
6321:
6314:
6307:
6299:
6292:
6291:
6284:
6277:
6267:
6260:
6248:
6247:
6240:
6234:
6229:
6228:
6225:
6224:
6217:
6216:
6209:
6208:
6205:
6199:
6196:
6191:
6186:
6180:
6179:
6172:
6171:
6168:
6163:
6158:
6153:
6147:
6146:
6139:
6138:
6135:
6130:
6125:
6118:
6116:
6110:
6109:
6102:
6101:
6098:
6093:
6088:
6083:
6073:
6072:
6065:
6060:
6057:
6056:
6049:
6048:
6041:
6031:
6024:
6016:
6015:
6005:
5998:
5991:
5983:
5982:
5975:
5968:
5961:
5950:
5949:
5942:
5935:
5925:
5913:
5912:
5905:
5900:
5896:
5894:
5891:
5890:
5870:
5865:
5864:
5859:
5854:
5852:
5849:
5848:
5846:
5826:
5825:
5820:
5815:
5810:
5805:
5799:
5798:
5793:
5788:
5783:
5778:
5772:
5771:
5766:
5761:
5756:
5751:
5745:
5744:
5739:
5734:
5729:
5724:
5714:
5713:
5705:
5703:
5700:
5699:
5673:
5670:
5669:
5667:
5664:
5639:
5636:
5635:
5633:
5632:coincides with
5614:
5609:
5608:
5600:
5592:
5590:
5587:
5586:
5584:
5554:
5551:
5550:
5548:
5530:
5528:
5525:
5524:
5522:
5492:
5489:
5488:
5486:
5467:
5462:
5461:
5453:
5451:
5448:
5447:
5445:
5418:
5415:
5414:
5412:
5394:
5392:
5389:
5388:
5386:
5361:
5356:
5355:
5353:
5350:
5349:
5347:
5328:
5323:
5322:
5320:
5317:
5316:
5314:
5296:
5293:
5292:
5290:
5271:
5266:
5265:
5263:
5260:
5259:
5257:
5230:
5227:
5226:
5224:
5206:
5203:
5202:
5200:
5173:
5170:
5169:
5167:
5141:
5136:
5135:
5133:
5130:
5129:
5127:
5105:
5102:
5101:
5099:
5098:The linear map
5077:
5072:
5071:
5069:
5066:
5065:
5063:
5045:
5042:
5041:
5039:
5018:
5015:
5014:
5012:
4990:
4986:
4984:
4981:
4980:
4978:
4960:
4957:
4956:
4954:
4935:
4931:
4929:
4926:
4925:
4923:
4905:
4902:
4901:
4899:
4881:
4878:
4877:
4875:
4856:
4852:
4850:
4847:
4846:
4844:
4825:
4821:
4819:
4816:
4815:
4813:
4794:
4790:
4781:
4777:
4769:
4766:
4765:
4763:
4719:
4716:
4715:
4713:
4685:
4680:
4679:
4671:
4668:
4667:
4665:
4644:
4642:
4639:
4638:
4636:
4618:
4615:
4614:
4612:
4593:
4589:
4587:
4584:
4583:
4581:
4519:
4514:
4513:
4507:
4503:
4491:
4486:
4485:
4477:
4474:
4473:
4452:
4451:
4443:
4436:
4430:
4429:
4423:
4419:
4412:
4406:
4402:
4398:
4394:
4386:
4383:
4382:
4371:scalar products
4351:
4347:
4345:
4342:
4341:
4339:
4321:
4319:
4316:
4315:
4313:
4294:
4289:
4288:
4273:
4268:
4267:
4265:
4262:
4261:
4259:
4240:
4236:
4234:
4231:
4230:
4228:
4206:
4204:
4201:
4200:
4198:
4179:
4174:
4173:
4158:
4153:
4152:
4150:
4147:
4146:
4144:
4126:
4124:
4121:
4120:
4118:
4100:
4098:
4095:
4094:
4092:
4089:
4065:
4063:
4060:
4059:
4057:
4039:
4037:
4034:
4033:
4031:
4007:
4004:
4003:
4001:
3980:
3978:
3975:
3974:
3972:
3949:
3944:
3943:
3941:
3938:
3937:
3935:
3917:
3915:
3912:
3911:
3909:
3891:
3888:
3887:
3885:
3863:
3858:
3857:
3855:
3852:
3851:
3849:
3827:
3825:
3822:
3821:
3819:
3795:
3792:
3791:
3789:
3768:
3766:
3763:
3762:
3760:
3738:
3736:
3733:
3732:
3730:
3712:
3709:
3708:
3706:
3702:
3671:
3669:
3666:
3665:
3663:
3642:
3640:
3637:
3636:
3634:
3616:
3614:
3611:
3610:
3608:
3590:
3588:
3585:
3584:
3582:
3557:
3555:
3552:
3551:
3549:
3526:
3521:
3520:
3518:
3515:
3514:
3512:
3491:
3489:
3486:
3485:
3483:
3462:
3457:
3456:
3454:
3451:
3450:
3448:
3428:
3426:
3423:
3422:
3420:
3399:
3397:
3394:
3393:
3391:
3372:
3367:
3366:
3364:
3361:
3360:
3358:
3336:
3331:
3330:
3324:
3320:
3318:
3315:
3314:
3312:
3293:
3288:
3287:
3285:
3282:
3281:
3279:
3261:
3259:
3256:
3255:
3253:
3226:
3221:
3220:
3218:
3215:
3214:
3212:
3194:
3192:
3189:
3188:
3186:
3183:
3178:
3174:
3173:The columns of
3148:
3145:
3144:
3142:
3120:
3117:
3116:
3114:
3090:
3087:
3086:
3084:
3062:
3059:
3058:
3056:
3029:
3026:
3025:
3023:
3020:Euclidean space
3001:
2998:
2997:
2995:
2988:singular values
2984:
2958:
2953:
2952:
2950:
2947:
2946:
2944:
2925:
2920:
2919:
2917:
2914:
2913:
2911:
2878:
2875:
2874:
2872:
2851:
2849:
2846:
2845:
2843:
2821:
2816:
2815:
2813:
2810:
2809:
2807:
2788:
2783:
2782:
2780:
2777:
2776:
2774:
2755:
2750:
2749:
2747:
2744:
2743:
2741:
2723:
2721:
2718:
2717:
2715:
2693:
2688:
2687:
2685:
2682:
2681:
2679:
2660:
2655:
2654:
2652:
2649:
2648:
2646:
2619:
2616:
2615:
2613:
2589:
2586:
2585:
2583:
2565:
2563:
2560:
2559:
2557:
2534:
2529:
2528:
2526:
2523:
2522:
2520:
2502:
2500:
2497:
2496:
2494:
2476:
2474:
2471:
2470:
2468:
2446:
2444:
2441:
2440:
2438:
2436:
2416:
2414:
2411:
2410:
2408:
2406:
2382:
2377:
2376:
2374:
2371:
2370:
2368:
2366:
2363:transformations
2339:
2334:
2333:
2331:
2328:
2327:
2325:
2303:
2299:
2297:
2294:
2293:
2291:
2272:
2267:
2266:
2264:
2261:
2260:
2258:
2253:represents the
2236:
2234:
2231:
2230:
2228:
2201:
2196:
2195:
2193:
2190:
2189:
2187:
2169:
2167:
2164:
2163:
2161:
2139:
2134:
2133:
2131:
2128:
2127:
2125:
2104:
2096:
2094:
2091:
2090:
2088:
2064:
2062:
2059:
2058:
2056:
2028:
2025:
2024:
2022:
2003:
1998:
1997:
1995:
1992:
1991:
1989:
1971:
1969:
1966:
1965:
1963:
1962:, the matrices
1935:
1932:
1931:
1929:
1911:
1909:
1906:
1905:
1903:
1900:
1883:
1879:
1876:
1872:
1866:
1863:singular values
1855:
1849:
1846:
1840:
1834:
1818:
1816:
1813:
1812:
1803:
1794:
1784:
1770:
1768:shearing matrix
1760:
1752:process control
1701:
1696:
1695:
1687:
1681:
1676:
1675:
1667:
1661:
1656:
1655:
1653:
1650:
1649:
1622:
1619:
1618:
1616:
1600:
1598:
1595:
1594:
1567:
1564:
1563:
1561:
1543:
1541:
1538:
1537:
1535:
1514:
1512:
1509:
1508:
1506:
1505:is the rank of
1467:
1464:
1463:
1461:
1434:
1431:
1430:
1428:
1410:
1408:
1405:
1404:
1402:
1383:
1372:
1371:
1363:
1361:
1358:
1357:
1355:
1328:
1326:
1323:
1322:
1320:
1302:
1300:
1297:
1296:
1294:
1276:
1274:
1271:
1270:
1268:
1252:
1250:
1247:
1246:
1226:
1222:
1220:
1217:
1216:
1193:
1191:
1188:
1187:
1185:
1184:is the rank of
1148:
1145:
1144:
1122:
1117:
1112:
1105:
1100:
1099:
1093:
1089:
1083:
1072:
1060:
1058:
1055:
1054:
1035:
1031:
1029:
1026:
1025:
1003:
998:
997:
982:
977:
976:
974:
971:
970:
968:
949:
944:
943:
928:
923:
922:
920:
917:
916:
914:
893:
891:
888:
887:
885:
867:
865:
862:
861:
859:
841:
839:
836:
835:
833:
815:
813:
810:
809:
807:
785:
783:
780:
779:
777:
774:singular values
755:
753:
750:
749:
747:
731:
729:
726:
725:
705:
701:
692:
688:
686:
683:
682:
659:
658:
653:
652:
647:
642:
640:
637:
636:
614:
612:
609:
608:
606:
588:
586:
583:
582:
580:
562:
560:
557:
556:
554:
536:
534:
531:
530:
528:
507:
502:
501:
499:
496:
495:
473:
470:
469:
451:
449:
446:
445:
443:
418:
415:
414:
398:
396:
393:
392:
364:
361:
360:
358:
340:
338:
335:
334:
332:
311:
307:
300:
292:
290:
287:
286:
268:
266:
263:
262:
260:
259:complex matrix
238:
235:
234:
203:
200:
199:
197:
157:
149:
140:
134:
131:
125:
119:
110:
104:
101:
95:
89:
83:
74:
68:
65:
59:
55:
49:
36:
32:
26:
17:
12:
11:
5:
32431:
32421:
32420:
32415:
32410:
32405:
32400:
32398:Linear algebra
32395:
32378:
32377:
32375:
32374:
32369:
32364:
32359:
32354:
32349:
32344:
32339:
32334:
32329:
32324:
32319:
32314:
32309:
32304:
32299:
32289:
32287:Corona theorem
32284:
32278:
32276:
32272:
32271:
32269:
32268:
32266:Wiener algebra
32262:
32260:
32256:
32255:
32253:
32252:
32247:
32242:
32237:
32232:
32227:
32222:
32217:
32212:
32206:
32204:
32200:
32199:
32197:
32196:
32186:
32184:Pseudospectrum
32181:
32176:
32174:Dirac spectrum
32170:
32168:
32164:
32163:
32161:
32160:
32155:
32150:
32145:
32139:
32137:
32133:
32132:
32130:
32129:
32128:
32127:
32117:
32112:
32107:
32102:
32097:
32091:
32085:
32083:
32079:
32078:
32076:
32075:
32070:
32065:
32060:
32055:
32050:
32045:
32040:
32035:
32029:
32027:
32023:
32022:
32020:
32019:
32014:
32009:
32004:
31999:
31994:
31993:
31992:
31987:
31982:
31971:
31969:
31965:
31964:
31962:
31961:
31956:
31951:
31946:
31941:
31936:
31930:
31928:
31922:
31921:
31919:
31918:
31913:
31905:
31897:
31889:
31883:
31881:
31877:
31876:
31874:
31873:
31868:
31863:
31858:
31853:
31848:
31843:
31837:
31835:
31831:
31830:
31828:
31827:
31825:Operator space
31822:
31817:
31812:
31807:
31802:
31797:
31792:
31787:
31785:Banach algebra
31782:
31776:
31774:
31773:Basic concepts
31770:
31769:
31758:
31757:
31750:
31743:
31735:
31726:
31725:
31723:
31722:
31711:
31708:
31707:
31705:
31704:
31699:
31694:
31689:
31687:Choquet theory
31684:
31679:
31673:
31671:
31667:
31666:
31664:
31663:
31653:
31648:
31643:
31638:
31633:
31628:
31623:
31618:
31613:
31608:
31603:
31597:
31595:
31591:
31590:
31588:
31587:
31582:
31576:
31574:
31570:
31569:
31567:
31566:
31561:
31556:
31551:
31546:
31541:
31539:Banach algebra
31535:
31533:
31529:
31528:
31526:
31525:
31520:
31515:
31510:
31505:
31500:
31495:
31490:
31485:
31480:
31474:
31472:
31468:
31467:
31465:
31464:
31462:Banach–Alaoglu
31459:
31454:
31449:
31444:
31439:
31434:
31429:
31424:
31418:
31416:
31410:
31409:
31406:
31405:
31403:
31402:
31397:
31392:
31390:Locally convex
31387:
31373:
31368:
31362:
31360:
31356:
31355:
31353:
31352:
31347:
31342:
31337:
31332:
31327:
31322:
31317:
31312:
31307:
31301:
31295:
31291:
31290:
31274:
31273:
31266:
31259:
31251:
31242:
31241:
31239:
31238:
31233:
31228:
31223:
31218:
31213:
31207:
31205:
31201:
31200:
31198:
31197:
31192:
31187:
31182:
31177:
31171:
31169:
31165:
31164:
31162:
31161:
31156:
31151:
31141:
31136:
31130:
31128:
31124:
31123:
31121:
31120:
31115:
31113:Floating point
31109:
31107:
31103:
31102:
31095:
31094:
31087:
31080:
31072:
31066:
31065:
31058:
31057:External links
31055:
31054:
31053:
31048:
31031:
31019:Rocha, Luis M.
31014:
30985:10.1.1.23.1831
30978:(4): 551–566.
30963:
30957:
30944:
30938:
30921:
30915:
30896:
30890:
30877:
30859:(4): 534–553.
30848:
30841:
30828:
30822:
30806:Golub, Gene H.
30802:
30776:(2): 205–224.
30766:Kahan, William
30762:Golub, Gene H.
30758:
30745:10.1.1.48.3740
30738:(5): 873–912.
30728:Kahan, William
30720:
30714:
30698:
30662:(4): 837–843.
30647:
30641:
30628:
30624:978-1420095388
30623:
30608:
30605:
30602:
30601:
30574:(5): 403–420.
30551:
30538:
30492:
30452:
30422:
30363:
30326:(4): 837–843.
30306:
30299:
30267:
30256:
30245:
30234:
30215:
30202:
30199:
30196:
30193:
30190:
30187:
30184:
30181:
30178:
30156:
30152:
30146:
30141:
30135:
30131:
30128:
30125:
30120:
30115:
30109:
30103:
30098:
30091:
30086:
30081:
30076:
30073:
30070:
30054:
30027:(1): 620–646.
30009:
29982:(2): 153–178.
29966:
29937:(3): 597–616.
29921:
29884:(3): 454–464.
29878:Expert Systems
29868:
29859:|journal=
29811:
29802:|journal=
29774:Riedl, John T.
29760:
29707:
29674:
29641:
29597:
29554:(11): e78913.
29534:
29475:
29416:
29357:
29335:
29307:
29276:
29265:
29246:(3): 1220–34.
29230:
29186:
29185:
29183:
29180:
29178:
29177:
29172:
29167:
29161:
29156:
29154:Singular value
29151:
29146:
29141:
29135:
29130:
29125:
29120:
29114:
29109:
29104:
29099:
29094:
29089:
29084:
29079:
29074:
29069:
29064:
29059:
29053:
29048:
29043:
29038:
29033:
29027:
29025:Canonical form
29022:
29016:
29014:
29011:
28954:
28950:
28930:Erhard Schmidt
28919:principal axis
28888:
28884:
28872:of the matrix
28854:Camille Jordan
28837:
28834:
28833:
28832:
28817:
28811:
28806:
28779:
28742:
28738:
28713:
28708:
28704:
28699:
28695:
28671:
28666:
28662:
28657:
28653:
28630:
28626:
28599:
28595:
28570:
28567:
28543:
28538:
28534:
28529:
28524:
28518:
28514:
28509:
28505:
28502:
28498:
28492:
28488:
28482:
28478:
28474:
28469:
28465:
28460:
28455:
28449:
28445:
28440:
28436:
28433:
28428:
28423:
28416:
28412:
28407:
28402:
28396:
28392:
28388:
28385:
28380:
28375:
28368:
28364:
28359:
28355:
28352:
28348:
28324:
28321:
28297:
28273:
28268:
28264:
28260:
28236:
28231:
28227:
28223:
28197:
28193:
28167:
28161:
28156:
28129:
28104:
28080:
28062:
28059:
28042:
28037:
28030:
28026:
28021:
27994:
27989:
27983:
27956:
27951:
27944:
27940:
27935:
27931:
27926:
27921:
27915:
27911:
27907:
27873:
27865:
27861:
27857:
27856:
27851:
27847:
27843:
27842:
27840:
27812:
27808:
27781:
27777:
27751:
27718:
27714:
27708:
27703:
27675:
27670:
27663:
27659:
27654:
27627:
27624:
27621:
27618:
27615:
27612:
27607:
27603:
27577:
27549:
27545:
27518:
27513:
27506:
27502:
27497:
27493:
27489:
27464:
27440:
27437:
27434:
27431:
27428:
27425:
27401:
27397:
27372:
27368:
27353:there exist a
27340:
27336:
27311:
27308:
27283:
27253:
27248:
27242:
27237:
27233:
27229:
27214:
27211:
27196:
27192:
27167:
27163:
27137:
27112:
27108:
27103:
27098:
27073:
27069:
27041:
27037:
27012:
27008:
26982:
26957:
26953:
26948:
26943:
26918:
26914:
26888:
26873:
26870:
26868:
26865:
26846:
26839:
26834:
26827:
26824:
26820:
26815:
26809:
26806:
26802:
26793:
26762:
26757:
26754:
26750:
26746:
26743:
26739:
26714:
26710:
26687:Frobenius norm
26671:
26667:
26640:
26636:
26607:
26602:
26598:
26592:
26588:
26579:
26569:
26566:
26562:
26558:
26534:
26528:
26523:
26517:
26512:
26506:
26502:
26499:
26494:
26489:
26485:
26481:
26477:
26473:
26468:
26465:
26461:
26457:
26433:
26429:
26424:
26418:
26413:
26407:
26403:
26400:
26397:
26394:
26390:
26386:
26382:
26378:
26354:
26351:
26348:
26330:
26327:
26311:
26305:
26300:
26274:
26270:
26266:
26262:
26257:
26251:
26246:
26241:
26238:
26235:
26232:
26229:
26225:
26221:
26193:
26189:
26164:
26159:
26155:
26151:
26147:
26142:
26136:
26131:
26126:
26102:
26098:
26094:
26090:
26085:
26079:
26074:
26069:
26041:
26037:
26033:
26029:
26024:
26018:
26013:
26008:
25981:
25978:
25973:
25968:
25962:
25957:
25952:
25949:
25946:
25942:
25938:
25913:
25887:
25883:
25860:
25855:
25851:
25824:
25820:
25794:
25763:
25759:
25734:
25702:
25677:
25663:
25660:
25658:
25655:
25632:
25607:
25603:
25591:of the matrix
25571:
25568:
25565:
25562:
25536:
25531:
25505:
25479:
25474:
25448:
25423:
25398:
25374:
25371:
25368:
25365:
25339:
25334:
25329:
25316:diagonal, and
25303:
25300:
25297:
25271:
25266:
25241:
25238:
25235:
25232:
25206:
25201:
25175:
25170:
25165:
25160:
25153:
25148:
25141:
25136:
25131:
25125:
25121:
25094:
25067:
25063:
25032:
25028:
25003:
25000:
24997:
24973:
24959:
24956:
24942:
24939:
24936:
24933:
24907:
24902:
24897:
24884:diagonal, and
24871:
24868:
24865:
24839:
24834:
24809:
24806:
24803:
24800:
24774:
24769:
24743:
24740:
24737:
24734:
24731:
24728:
24725:
24722:
24719:
24693:
24688:
24661:
24634:
24629:
24601:
24596:
24570:
24545:
24520:
24496:
24491:
24486:
24481:
24474:
24469:
24462:
24457:
24452:
24448:
24422:
24407:
24404:
24389:
24385:
24356:
24353:
24350:
24347:
24344:
24341:
24338:
24335:
24332:
24306:
24303:
24300:
24297:
24271:
24266:
24261:
24248:diagonal, and
24235:
24232:
24229:
24203:
24198:
24173:
24170:
24167:
24164:
24138:
24133:
24107:
24103:
24078:
24052:
24047:
24021:
24017:
23991:
23966:
23940:
23935:
23907:
23902:
23878:
23875:
23872:
23869:
23866:
23863:
23860:
23857:
23854:
23832:
23827:
23822:
23817:
23810:
23805:
23798:
23793:
23788:
23784:
23758:
23743:
23740:
23725:
23700:
23675:
23672:
23669:
23610:
23607:
23585:
23581:
23575:
23570:
23566:
23560:
23556:
23552:
23549:
23546:
23543:
23538:
23534:
23528:
23523:
23519:
23513:
23509:
23505:
23502:
23499:
23496:
23491:
23487:
23481:
23476:
23472:
23466:
23462:
23458:
23455:
23452:
23449:
23444:
23440:
23434:
23429:
23425:
23419:
23415:
23411:
23408:
23405:
23402:
23397:
23393:
23387:
23382:
23378:
23372:
23368:
23364:
23361:
23358:
23355:
23350:
23346:
23340:
23335:
23331:
23325:
23321:
23317:
23314:
23311:
23306:
23303:
23298:
23293:
23286:
23282:
23277:
23273:
23268:
23263:
23256:
23252:
23247:
23243:
23238:
23233:
23226:
23222:
23217:
23213:
23208:
23203:
23196:
23192:
23187:
23181:
23178:
23176:
23174:
23165:
23160:
23153:
23148:
23144:
23140:
23135:
23130:
23126:
23122:
23117:
23112:
23108:
23104:
23099:
23094:
23090:
23085:
23081:
23076:
23070:
23062:
23057:
23050:
23046:
23041:
23037:
23032:
23027:
23020:
23016:
23011:
23007:
23002:
22997:
22990:
22986:
22981:
22977:
22972:
22967:
22960:
22956:
22951:
22945:
22938:
22933:
22928:
22921:
22917:
22912:
22908:
22903:
22898:
22891:
22887:
22882:
22878:
22873:
22868:
22861:
22857:
22852:
22848:
22843:
22838:
22831:
22827:
22822:
22816:
22813:
22811:
22807:
22803:
22799:
22798:
22785:Pauli matrices
22770:
22766:
22742:
22718:
22714:
22709:
22705:
22680:
22676:
22670:
22666:
22662:
22657:
22653:
22647:
22643:
22639:
22634:
22630:
22624:
22620:
22616:
22612:
22606:
22602:
22598:
22594:
22571:
22568:
22565:
22551:
22548:
22523:
22519:
22515:
22490:
22485:
22480:
22474:
22469:
22465:
22461:
22436:
22431:
22426:
22420:
22416:
22412:
22386:
22356:
22351:
22347:
22343:
22298:
22293:
22286:
22282:
22278:
22273:
22268:
22267:
22263:
22259:
22256:
22252:
22251:
22249:
22223:
22219:
22213:
22208:
22183:
22178:
22173:
22167:
22141:
22101:
22098:
22095:
22067:
22064:
22061:
21997:
21993:
21989:
21986:
21981:
21977:
21973:
21970:
21941:
21916:
21892:
21868:
21864:
21858:
21854:
21850:
21847:
21842:
21838:
21834:
21831:
21796:
21791:
21787:
21783:
21780:
21777:
21753:
21750:
21747:
21744:
21720:
21717:
21714:
21711:
21675:
21672:
21669:
21666:
21642:
21637:
21633:
21629:
21626:
21623:
21591:
21577:
21576:
21560:
21555:
21549:
21523:
21517:
21512:
21483:
21459:
21445:
21430:
21424:
21419:
21392:
21378:
21362:
21357:
21351:
21318:
21299:
21296:
21283:
21280:
21277:
21274:
21254:
21249:
21245:
21241:
21238:
21235:
21215:
21195:
21172:
21152:
21130:
21127:
21124:
21119:
21115:
21111:
21108:
21097:to zero them,
21078:
21058:
21055:
21042:
21022:
20998:
20978:
20956:
20952:
20948:
20945:
20942:
20939:
20916:
20896:
20893:
20890:
20887:
20884:
20881:
20878:
20875:
20872:
20869:
20866:
20863:
20860:
20857:
20854:
20834:
20831:
20828:
20825:
20822:
20802:
20782:
20762:
20759:
20756:
20753:
20750:
20747:
20744:
20741:
20721:
20699:
20696:
20693:
20690:
20687:
20684:
20681:
20678:
20675:
20672:
20669:
20666:
20648:
20645:
20643:
20640:
20635:respectively.
20622:
20617:
20612:
20584:
20579:
20552:
20526:
20501:
20497:
20493:
20489:
20485:
20482:
20451:
20446:
20441:
20434:
20430:
20426:
20423:
20421:
20417:
20412:
20405:
20400:
20395:
20394:
20391:
20386:
20381:
20374:
20370:
20366:
20363:
20361:
20357:
20352:
20346:
20342:
20341:
20317:
20312:
20308:
20304:
20301:
20296:
20292:
20288:
20285:
20280:
20276:
20253:
20250:
20247:
20243:
20239:
20236:
20233:
20229:
20225:
20194:
20189:
20154:
20149:
20119:
20114:
20109:
20102:
20098:
20094:
20091:
20088:
20085:
20082:
20080:
20076:
20071:
20064:
20059:
20054:
20053:
20050:
20047:
20044:
20039:
20034:
20027:
20023:
20019:
20016:
20013:
20011:
20007:
20002:
19996:
19992:
19991:
19966:
19960:
19955:
19950:
19947:
19942:
19938:
19934:
19930:
19924:
19919:
19914:
19911:
19906:
19902:
19898:
19894:
19889:
19883:
19878:
19873:
19870:
19867:
19864:
19835:
19830:
19826:
19800:
19773:
19768:
19740:
19735:
19702:
19697:
19669:
19664:
19638:
19634:
19630:
19626:
19622:
19619:
19593:
19589:
19564:
19559:
19554:
19526:
19521:
19493:
19489:
19464:
19459:
19456:
19453:
19449:
19423:
19396:
19393:
19390:
19386:
19360:
19335:
19301:
19298:
19295:
19291:
19264:
19261:
19258:
19254:
19229:
19224:
19221:
19218:
19214:
19210:
19205:
19202:
19199:
19195:
19183:restricted to
19170:
19146:
19141:
19138:
19135:
19131:
19127:
19123:
19103:
19098:
19095:
19092:
19088:
19084:
19080:
19060:
19056:
19051:
19045:
19040:
19035:
19032:
19028:
19024:
19020:
19016:
19013:
18991:
18986:
18964:
18961:
18958:
18955:
18952:
18928:
18925:
18922:
18918:
18893:
18890:
18887:
18862:
18833:
18807:
18782:
18778:
18752:
18727:
18723:
18718:
18711:
18705:
18689:
18686:
18670:
18664:
18659:
18631:
18626:
18620:
18594:
18588:
18583:
18555:
18550:
18544:
18519:
18514:
18509:
18503:
18498:
18494:
18490:
18465:
18461:
18457:
18452:
18447:
18442:
18434:
18431:
18425:
18418:
18413:
18408:
18403:
18397:
18394:
18393:
18388:
18383:
18378:
18370:
18367:
18361:
18356:
18355:
18353:
18346:
18338:
18333:
18328:
18324:
18319:
18314:
18313:
18311:
18306:
18301:
18295:
18287:
18282:
18277:
18273:
18268:
18263:
18262:
18260:
18252:
18246:
18243:
18242:
18237:
18231:
18228:
18226:
18223:
18222:
18219:
18216:
18211:
18208:
18203:
18197:
18196:
18194:
18189:
18188:
18186:
18179:
18171:
18166:
18161:
18157:
18152:
18147:
18146:
18144:
18120:
18117:
18114:
18093:
18070:
18065:
18060:
18030:
18025:
18019:
18016:
18015:
18010:
18004:
18001:
17999:
17996:
17995:
17992:
17989:
17984:
17981:
17975:
17970:
17969:
17967:
17962:
17961:
17959:
17954:
17950:
17923:
17918:
17890:
17885:
17856:
17848:
17843:
17838:
17834:
17829:
17824:
17823:
17821:
17816:
17812:
17789:
17784:
17760:
17755:
17731:
17725:
17721:
17716:
17709:
17706:
17701:
17696:
17690:
17682:
17679:
17674:
17669:
17664:
17657:
17654:
17649:
17644:
17637:
17632:
17626:
17620:
17615:
17608:
17603:
17598:
17589:
17586:
17581:
17576:
17571:
17566:
17561:
17554:
17549:
17544:
17531:. Also, since
17520:
17500:
17479:
17456:
17451:
17427:
17422:
17398:
17393:
17366:
17361:
17358:
17355:
17346:
17335:
17328:
17323:
17317:
17311:
17307:
17303:
17300:
17295:
17291:
17285:
17261:
17256:
17233:
17227:
17222:
17198:
17193:
17169:
17164:
17161:
17158:
17154:
17148:
17143:
17138:
17116:
17111:
17083:
17078:
17075:
17072:
17063:
17052:
17045:
17040:
17034:
17028:
17024:
17020:
17017:
17012:
17008:
17002:
16978:
16973:
16970:
16967:
16963:
16957:
16952:
16946:
16942:
16920:
16915:
16912:
16909:
16905:
16899:
16895:
16891:
16870:
16864:
16859:
16835:
16830:
16827:
16824:
16820:
16814:
16809:
16804:
16783:
16779:
16774:
16769:
16764:
16757:
16752:
16746:
16725:
16721:
16717:
16712:
16707:
16701:
16678:
16674:
16670:
16665:
16660:
16655:
16650:
16645:
16640:
16634:
16630:
16627:
16623:
16619:
16616:
16611:
16606:
16601:
16594:
16589:
16584:
16580:
16576:
16572:
16568:
16563:
16558:
16553:
16545:
16542:
16536:
16527:
16524:
16519:
16514:
16507:
16502:
16496:
16492:
16487:
16482:
16477:
16469:
16466:
16460:
16453:
16448:
16422:
16415:
16412:
16407:
16402:
16395:
16390:
16384:
16380:
16375:
16370:
16337:
16332:
16327:
16322:
16319:
16317:
16313:
16308:
16303:
16296:
16291:
16286:
16281:
16276:
16271:
16264:
16259:
16254:
16253:
16250:
16245:
16240:
16235:
16232:
16230:
16226:
16221:
16214:
16209:
16204:
16199:
16198:
16195:
16190:
16185:
16180:
16177:
16175:
16171:
16166:
16159:
16154:
16149:
16144:
16143:
16119:
16114:
16090:
16085:
16062:
16040:
16036:
16031:
16026:
16020:
15997:
15993:
15989:
15984:
15979:
15973:
15967:
15962:
15955:
15950:
15945:
15939:
15935:
15931:
15926:
15921:
15915:
15909:
15904:
15897:
15892:
15887:
15861:
15856:
15850:
15847:
15845:
15842:
15841:
15838:
15835:
15832:
15828:
15827:
15825:
15820:
15815:
15807:
15802:
15796:
15790:
15785:
15778:
15773:
15768:
15763:
15759:
15754:
15748:
15742:
15737:
15730:
15725:
15720:
15715:
15714:
15709:
15704:
15698:
15692:
15687:
15680:
15675:
15670:
15665:
15661:
15656:
15650:
15644:
15639:
15632:
15627:
15622:
15617:
15616:
15614:
15609:
15604:
15596:
15591:
15584:
15580:
15575:
15570:
15569:
15567:
15560:
15554:
15549:
15542:
15534:
15529:
15524:
15519:
15518:
15513:
15508:
15503:
15498:
15497:
15495:
15470:
15448:
15442:
15437:
15413:
15408:
15384:
15379:
15355:
15347:
15342:
15337:
15333:
15328:
15323:
15322:
15320:
15315:
15311:
15289:
15268:
15265:
15260:
15257:
15250:
15246:
15220:
15214:
15209:
15187:
15184:
15181:
15160:
15150:-th column of
15139:
15117:
15114:
15107:
15103:
15077:
15071:
15066:
15044:
15024:
15003:
14982:
14979:
14976:
14973:
14970:
14967:
14964:
14961:
14940:
14934:
14929:
14907:
14887:
14884:
14881:
14860:
14837:
14832:
14826:
14823:
14821:
14818:
14817:
14814:
14811:
14808:
14804:
14803:
14801:
14796:
14790:
14786:
14779:
14775:
14770:
14764:
14759:
14752:
14747:
14722:
14699:
14696:
14693:
14666:
14660:
14655:
14631:
14628:
14625:
14602:
14589:
14586:
14567:
14539:
14536:
14511:
14506:
14500:
14495:
14490:
14487:
14483:
14479:
14476:
14451:
14426:
14422:
14397:
14373:
14370:
14366:
14362:
14359:
14334:
14308:
14283:
14279:
14253:
14228:
14224:
14220:
14217:
14213:
14208:
14183:
14179:
14175:
14171:
14167:
14164:
14160:
14156:
14153:
14150:
14146:
14140:
14135:
14130:
14127:
14124:
14121:
14117:
14112:
14106:
14101:
14096:
14071:
14043:
14013:
13989:
13986:
13962:
13959:
13955:
13949:
13944:
13939:
13936:
13933:
13930:
13926:
13921:
13915:
13910:
13905:
13880:
13853:
13850:
13847:
13844:
13841:
13837:
13833:
13830:
13807:
13777:
13770:
13765:
13759:
13754:
13749:
13746:
13744:
13741:
13737:
13736:
13732:
13728:
13725:
13723:
13719:
13714:
13709:
13708:
13705:
13701:
13698:
13671:
13668:
13665:
13640:
13610:
13585:
13581:
13577:
13574:
13570:
13565:
13539:
13514:
13503:An eigenvalue
13500:
13497:
13423:
13381:
13376:
13366:
13362:
13359:
13336:modal analysis
13307:
13306:Other examples
13304:
13287:
13284:
13270:
13267:
13253:
13247:
13242:
13237:
13233:
13207:
13203:
13199:
13196:
13174:
13170:
13166:
13162:
13156:
13151:
13137:
13133:
13128:
13124:
13120:
13115:
13111:
13106:
13103:
13098:
13094:
13082:Specifically,
13069:
13065:
13039:
13013:
12995:shape analysis
12978:
12973:
12969:
12964:
12960:
12955:
12951:
12947:
12922:
12917:
12911:
12907:
12903:
12876:
12871:
12865:
12861:
12857:
12830:
12825:
12819:
12814:
12810:
12806:
12780:
12753:
12748:
12742:
12737:
12712:
12707:
12702:
12696:
12671:
12667:
12663:
12659:
12645:Frobenius norm
12630:
12626:
12600:
12571:
12556:
12553:
12538:
12530:
12525:
12521:
12515:
12511:
12504:
12499:
12495:
12489:
12486:
12461:
12435:
12399:
12395:
12368:
12363:
12335:
12331:
12306:
12280:
12275:
12247:
12242:
12216:
12211:
12206:
12201:
12196:
12191:
12184:
12180:
12174:
12170:
12166:
12161:
12156:
12149:
12145:
12141:
12137:
12111:
12086:
12081:
12077:
12071:
12067:
12063:
12058:
12055:
12051:
12026:
12022:
12018:
12014:
12010:
12006:
11976:
11961:
11958:
11933:
11910:
11886:
11882:
11856:
11851:
11846:
11838:
11834:
11826:
11822:
11816:
11812:
11784:
11780:
11757:
11754:
11751:
11746:
11738:
11734:
11725:
11720:
11717:
11692:
11688:
11659:
11645:Frobenius norm
11630:
11601:
11597:
11570:
11551:
11548:
11544:rounding error
11540:effective rank
11526:
11502:
11471:
11467:
11440:
11435:
11405:
11401:
11375:
11349:
11323:
11298:
11294:
11271:
11268:
11252:
11229:
11226:
11223:
11219:
11215:
11192:
11187:
11157:
11138:
11135:
11121:
11117:
11092:
11088:
11061:
11056:
11029:
11025:
11021:
11015:
11010:
10983:
10956:
10926:
10900:
10874:
10849:
10845:
10815:
10789:
10763:
10737:
10712:
10708:
10682:
10656:
10652:
10648:
10643:
10624:
10621:
10599:
10576:
10571:
10547:
10542:
10537:
10530:
10525:
10519:
10515:
10510:
10505:
10477:
10472:
10466:
10461:
10457:
10453:
10427:
10408:
10405:
10403:
10400:
10385:
10382:
10379:
10351:
10347:
10321:
10295:
10269:
10243:
10217:
10191:
10165:
10140:
10136:
10106:
10101:
10095:
10091:
10087:
10060:
10055:
10049:
10044:
10040:
10036:
10011:
10007:
10002:
9998:
9994:
9965:
9960:
9955:
9927:
9922:
9894:
9890:
9863:
9860:
9856:
9829:
9824:
9818:
9813:
9809:
9805:
9778:
9774:
9744:
9717:
9713:
9687:
9661:
9634:
9629:
9623:
9618:
9614:
9610:
9565:
9551:
9550:
9537:
9532:
9527:
9521:
9495:
9489:
9484:
9453:
9439:
9426:
9421:
9416:
9410:
9384:
9370:
9357:
9353:
9347:
9342:
9311:
9278:
9273:
9268:
9263:
9258:
9253:
9247:
9243:
9239:
9235:
9230:
9225:
9218:
9213:
9207:
9200:
9195:
9189:
9184:
9180:
9177:
9175:
9171:
9166:
9160:
9156:
9155:
9152:
9147:
9142:
9137:
9133:
9127:
9122:
9117:
9113:
9109:
9104:
9099:
9093:
9088:
9081:
9076:
9069:
9064:
9058:
9054:
9051:
9049:
9046:
9040:
9035:
9030:
9029:
9005:
9000:
8995:
8989:
8984:
8980:
8976:
8950:
8914:
8911:
8908:
8894:
8891:
8877:
8873:
8847:
8821:
8795:
8769:
8743:
8717:
8691:
8664:
8661:
8657:
8628:
8602:
8577:
8553:
8550:
8547:
8522:
8497:
8494:
8491:
8488:
8464:
8461:
8458:
8433:
8408:
8405:
8402:
8378:
8375:
8372:
8369:
8341:
8337:
8302:
8299:
8274:
8248:
8222:
8194:
8189:
8161:
8156:
8137:
8136:
8123:
8119:
8092:
8088:
8062:
8048:
8035:
8031:
8004:
8000:
7974:
7957:
7943:
7918:
7893:
7890:
7887:
7858:
7832:
7807:
7804:
7801:
7798:
7795:
7792:
7789:
7786:
7762:
7758:
7732:
7705:
7700:
7694:
7689:
7685:
7681:
7666:respectively.
7653:
7650:
7617:
7591:
7562:
7558:
7554:
7551:
7548:
7546:
7543:
7537:
7532:
7527:
7526:
7523:
7519:
7515:
7512:
7509:
7507:
7504:
7501:
7497:
7496:
7470:
7466:
7440:
7413:
7409:
7383:
7357:
7342:singular value
7326:
7312:
7309:
7307:
7304:
7287:
7278:
7272:
7267:
7261:
7258:
7254:
7251:
7247:
7242:
7237:
7233:
7232:
7226:
7221:
7217:
7212:
7206:
7203:
7199:
7196:
7192:
7187:
7181:
7180:
7174:
7168:
7165:
7161:
7158:
7154:
7151:
7147:
7142:
7136:
7135:
7131:
7127:
7124:
7120:
7117:
7113:
7110:
7106:
7103:
7099:
7098:
7094:
7090:
7087:
7083:
7080:
7076:
7073:
7069:
7066:
7062:
7061:
7059:
7054:
7049:
7044:
7017:
6986:
6981:
6976:
6971:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6941:
6938:
6935:
6933:
6930:
6928:
6925:
6923:
6920:
6918:
6915:
6914:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6887:
6884:
6881:
6879:
6876:
6874:
6871:
6869:
6866:
6864:
6861:
6860:
6857:
6854:
6852:
6849:
6847:
6844:
6842:
6839:
6837:
6834:
6833:
6831:
6826:
6823:
6821:
6817:
6812:
6806:
6802:
6801:
6796:
6791:
6786:
6781:
6775:
6772:
6770:
6767:
6765:
6762:
6760:
6757:
6756:
6753:
6750:
6748:
6745:
6743:
6740:
6738:
6735:
6734:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6712:
6709:
6706:
6704:
6701:
6699:
6696:
6694:
6691:
6690:
6688:
6683:
6680:
6678:
6674:
6669:
6663:
6659:
6658:
6628:
6623:
6596:
6561:
6556:
6529:
6503:
6472:
6463:
6457:
6454:
6450:
6447:
6443:
6440:
6436:
6431:
6426:
6422:
6421:
6417:
6413:
6410:
6406:
6403:
6399:
6396:
6392:
6389:
6385:
6384:
6380:
6376:
6373:
6369:
6366:
6362:
6359:
6356:
6352:
6349:
6345:
6344:
6338:
6333:
6329:
6326:
6322:
6319:
6315:
6312:
6308:
6303:
6298:
6294:
6293:
6289:
6285:
6282:
6278:
6275:
6272:
6268:
6265:
6261:
6258:
6254:
6253:
6251:
6246:
6243:
6241:
6237:
6232:
6227:
6226:
6221:
6212:
6206:
6202:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6181:
6175:
6169:
6167:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6148:
6142:
6136:
6134:
6131:
6129:
6126:
6122:
6117:
6115:
6112:
6111:
6105:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6078:
6076:
6071:
6068:
6066:
6063:
6059:
6058:
6053:
6046:
6042:
6039:
6036:
6032:
6029:
6025:
6022:
6018:
6017:
6013:
6010:
6006:
6003:
5999:
5996:
5992:
5989:
5985:
5984:
5980:
5976:
5973:
5969:
5966:
5962:
5959:
5956:
5952:
5951:
5947:
5943:
5940:
5936:
5933:
5930:
5926:
5923:
5919:
5918:
5916:
5911:
5908:
5906:
5903:
5899:
5898:
5873:
5868:
5862:
5857:
5830:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5800:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5773:
5770:
5767:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5746:
5743:
5740:
5738:
5735:
5733:
5730:
5728:
5725:
5723:
5720:
5719:
5717:
5712:
5708:
5683:
5680:
5677:
5663:
5660:
5646:
5643:
5617:
5612:
5607:
5603:
5599:
5595:
5570:
5567:
5564:
5561:
5558:
5533:
5508:
5505:
5502:
5499:
5496:
5470:
5465:
5460:
5456:
5431:
5428:
5425:
5422:
5397:
5369:
5364:
5359:
5331:
5326:
5300:
5274:
5269:
5243:
5240:
5237:
5234:
5210:
5186:
5183:
5180:
5177:
5149:
5144:
5139:
5109:
5085:
5080:
5075:
5049:
5022:
4998:
4993:
4989:
4964:
4938:
4934:
4909:
4885:
4859:
4855:
4828:
4824:
4797:
4793:
4789:
4784:
4780:
4776:
4773:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4699:
4696:
4693:
4688:
4683:
4678:
4675:
4651:
4647:
4622:
4596:
4592:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4527:
4522:
4517:
4510:
4506:
4502:
4499:
4494:
4489:
4484:
4481:
4456:
4450:
4446:
4442:
4439:
4437:
4435:
4432:
4431:
4426:
4422:
4418:
4415:
4413:
4409:
4405:
4401:
4400:
4397:
4393:
4390:
4354:
4350:
4324:
4297:
4292:
4287:
4284:
4281:
4276:
4271:
4243:
4239:
4209:
4182:
4177:
4172:
4169:
4166:
4161:
4156:
4129:
4103:
4088:
4085:
4084:
4083:
4068:
4042:
4017:
4014:
4011:
3998:
3983:
3952:
3947:
3920:
3895:
3882:
3866:
3861:
3830:
3805:
3802:
3799:
3786:
3771:
3741:
3716:
3701:
3698:
3690:diagonalizable
3674:
3662:However, when
3649:
3645:
3619:
3593:
3560:
3529:
3524:
3498:
3494:
3470:
3465:
3460:
3435:
3431:
3406:
3402:
3375:
3370:
3344:
3339:
3334:
3327:
3323:
3296:
3291:
3264:
3229:
3224:
3197:
3182:
3171:
3152:
3124:
3100:
3097:
3094:
3066:
3039:
3036:
3033:
3005:
2983:
2980:
2966:
2961:
2956:
2928:
2923:
2897:
2894:
2891:
2888:
2885:
2882:
2858:
2854:
2829:
2824:
2819:
2791:
2786:
2758:
2753:
2726:
2701:
2696:
2691:
2663:
2658:
2632:
2629:
2626:
2623:
2599:
2596:
2593:
2568:
2556:If the matrix
2537:
2532:
2505:
2479:
2449:
2419:
2385:
2380:
2342:
2337:
2311:
2306:
2302:
2290:by the factor
2275:
2270:
2239:
2204:
2199:
2172:
2147:
2142:
2137:
2110:
2107:
2103:
2099:
2071:
2067:
2038:
2035:
2032:
2006:
2001:
1974:
1945:
1942:
1939:
1914:
1899:
1896:
1881:
1874:
1853:
1844:
1839:. The lengths
1821:
1759:
1756:
1709:
1704:
1699:
1694:
1690:
1684:
1679:
1674:
1670:
1664:
1659:
1632:
1629:
1626:
1603:
1577:
1574:
1571:
1546:
1521:
1517:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1447:
1444:
1441:
1438:
1413:
1386:
1381:
1378:
1375:
1370:
1366:
1335:
1331:
1305:
1279:
1255:
1232:
1229:
1225:
1200:
1196:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1130:
1125:
1120:
1115:
1108:
1103:
1096:
1092:
1086:
1081:
1078:
1075:
1071:
1067:
1063:
1038:
1034:
1011:
1006:
1001:
996:
993:
990:
985:
980:
952:
947:
942:
939:
936:
931:
926:
896:
870:
844:
818:
788:
758:
734:
711:
708:
704:
700:
695:
691:
668:
662:
656:
650:
645:
617:
591:
579:is real, then
565:
539:
510:
505:
483:
480:
477:
454:
428:
425:
422:
401:
389:unitary matrix
374:
371:
368:
343:
320:
314:
310:
306:
303:
299:
295:
271:
248:
245:
242:
213:
210:
207:
163:linear algebra
156:
155:
148:The action of
143:
138:
129:
118:The action of
113:
108:
99:
82:The action of
77:
72:
63:
48:The action of
42:
15:
9:
6:
4:
3:
2:
32430:
32419:
32416:
32414:
32411:
32409:
32408:Matrix theory
32406:
32404:
32401:
32399:
32396:
32394:
32391:
32390:
32388:
32373:
32370:
32368:
32365:
32363:
32360:
32358:
32355:
32353:
32350:
32348:
32345:
32343:
32340:
32338:
32335:
32333:
32330:
32328:
32325:
32323:
32320:
32318:
32315:
32313:
32310:
32308:
32305:
32303:
32300:
32297:
32293:
32290:
32288:
32285:
32283:
32280:
32279:
32277:
32273:
32267:
32264:
32263:
32261:
32257:
32251:
32248:
32246:
32243:
32241:
32238:
32236:
32233:
32231:
32228:
32226:
32223:
32221:
32218:
32216:
32213:
32211:
32208:
32207:
32205:
32203:Miscellaneous
32201:
32194:
32190:
32187:
32185:
32182:
32180:
32177:
32175:
32172:
32171:
32169:
32165:
32159:
32156:
32154:
32151:
32149:
32146:
32144:
32141:
32140:
32138:
32134:
32126:
32123:
32122:
32121:
32118:
32116:
32113:
32111:
32108:
32106:
32103:
32101:
32098:
32096:
32092:
32090:
32087:
32086:
32084:
32080:
32074:
32071:
32069:
32066:
32064:
32061:
32059:
32056:
32054:
32051:
32049:
32046:
32044:
32041:
32039:
32036:
32034:
32031:
32030:
32028:
32024:
32018:
32015:
32013:
32010:
32008:
32005:
32003:
32000:
31998:
31995:
31991:
31988:
31986:
31983:
31981:
31978:
31977:
31976:
31973:
31972:
31970:
31968:Decomposition
31966:
31960:
31957:
31955:
31952:
31950:
31947:
31945:
31942:
31940:
31937:
31935:
31932:
31931:
31929:
31927:
31923:
31917:
31914:
31912:
31909:
31906:
31904:
31901:
31898:
31896:
31893:
31890:
31888:
31885:
31884:
31882:
31878:
31872:
31869:
31867:
31864:
31862:
31859:
31857:
31854:
31852:
31849:
31847:
31844:
31842:
31839:
31838:
31836:
31832:
31826:
31823:
31821:
31818:
31816:
31813:
31811:
31808:
31806:
31803:
31801:
31798:
31796:
31793:
31791:
31788:
31786:
31783:
31781:
31778:
31777:
31775:
31771:
31767:
31763:
31756:
31751:
31749:
31744:
31742:
31737:
31736:
31733:
31721:
31713:
31712:
31709:
31703:
31700:
31698:
31695:
31693:
31692:Weak topology
31690:
31688:
31685:
31683:
31680:
31678:
31675:
31674:
31672:
31668:
31661:
31657:
31654:
31652:
31649:
31647:
31644:
31642:
31639:
31637:
31634:
31632:
31629:
31627:
31624:
31622:
31619:
31617:
31616:Index theorem
31614:
31612:
31609:
31607:
31604:
31602:
31599:
31598:
31596:
31592:
31586:
31583:
31581:
31578:
31577:
31575:
31573:Open problems
31571:
31565:
31562:
31560:
31557:
31555:
31552:
31550:
31547:
31545:
31542:
31540:
31537:
31536:
31534:
31530:
31524:
31521:
31519:
31516:
31514:
31511:
31509:
31506:
31504:
31501:
31499:
31496:
31494:
31491:
31489:
31486:
31484:
31481:
31479:
31476:
31475:
31473:
31469:
31463:
31460:
31458:
31455:
31453:
31450:
31448:
31445:
31443:
31440:
31438:
31435:
31433:
31430:
31428:
31425:
31423:
31420:
31419:
31417:
31415:
31411:
31401:
31398:
31396:
31393:
31391:
31388:
31385:
31381:
31377:
31374:
31372:
31369:
31367:
31364:
31363:
31361:
31357:
31351:
31348:
31346:
31343:
31341:
31338:
31336:
31333:
31331:
31328:
31326:
31323:
31321:
31318:
31316:
31313:
31311:
31308:
31306:
31303:
31302:
31299:
31296:
31292:
31287:
31283:
31279:
31272:
31267:
31265:
31260:
31258:
31253:
31252:
31249:
31237:
31234:
31232:
31229:
31227:
31224:
31222:
31219:
31217:
31214:
31212:
31209:
31208:
31206:
31202:
31196:
31193:
31191:
31188:
31186:
31183:
31181:
31178:
31176:
31173:
31172:
31170:
31166:
31160:
31157:
31155:
31152:
31149:
31145:
31142:
31140:
31137:
31135:
31132:
31131:
31129:
31125:
31119:
31116:
31114:
31111:
31110:
31108:
31104:
31100:
31093:
31088:
31086:
31081:
31079:
31074:
31073:
31070:
31064:
31061:
31060:
31051:
31045:
31041:
31037:
31036:"Section 2.6"
31032:
31028:
31024:
31020:
31015:
31011:
31007:
31003:
30999:
30995:
30991:
30986:
30981:
30977:
30973:
30969:
30964:
30960:
30954:
30950:
30945:
30941:
30935:
30930:
30929:
30922:
30918:
30912:
30908:
30903:
30897:
30893:
30887:
30883:
30878:
30874:
30870:
30866:
30862:
30858:
30854:
30849:
30846:
30842:
30838:
30834:
30829:
30825:
30819:
30815:
30811:
30807:
30803:
30799:
30795:
30791:
30787:
30783:
30779:
30775:
30771:
30767:
30763:
30759:
30755:
30751:
30746:
30741:
30737:
30733:
30729:
30725:
30724:Demmel, James
30721:
30717:
30711:
30707:
30703:
30699:
30695:
30691:
30687:
30683:
30678:
30673:
30669:
30665:
30661:
30657:
30653:
30648:
30644:
30638:
30634:
30629:
30626:
30620:
30616:
30611:
30610:
30597:
30593:
30589:
30585:
30581:
30577:
30573:
30569:
30565:
30561:
30555:
30548:
30542:
30534:
30530:
30526:
30522:
30518:
30514:
30510:
30506:
30502:
30496:
30488:
30484:
30480:
30476:
30472:
30468:
30467:
30466:Psychometrika
30462:
30456:
30445:
30438:
30437:
30432:
30426:
30418:
30414:
30409:
30404:
30399:
30394:
30390:
30386:
30382:
30378:
30374:
30367:
30359:
30355:
30351:
30347:
30342:
30337:
30333:
30329:
30325:
30321:
30317:
30310:
30302:
30296:
30292:
30288:
30284:
30283:
30278:
30271:
30265:
30260:
30254:
30249:
30243:
30238:
30230:
30226:
30219:
30200:
30197:
30194:
30188:
30185:
30179:
30154:
30144:
30126:
30118:
30101:
30089:
30084:
30071:
30068:
30058:
30050:
30046:
30042:
30038:
30034:
30030:
30026:
30022:
30021:
30013:
30005:
30001:
29997:
29993:
29989:
29985:
29981:
29977:
29976:Astrodynamics
29970:
29962:
29958:
29954:
29950:
29945:
29940:
29936:
29932:
29925:
29917:
29913:
29909:
29905:
29901:
29897:
29892:
29887:
29883:
29879:
29872:
29864:
29851:
29843:
29839:
29834:
29829:
29822:
29815:
29807:
29794:
29786:
29779:
29775:
29771:
29764:
29756:
29752:
29748:
29744:
29740:
29736:
29731:
29726:
29723:(2): 024010.
29722:
29718:
29711:
29702:
29697:
29693:
29689:
29685:
29678:
29669:
29664:
29661:(1–3): 1–14.
29660:
29656:
29652:
29645:
29636:
29631:
29627:
29623:
29619:
29615:
29608:
29601:
29593:
29589:
29585:
29580:
29575:
29570:
29565:
29561:
29557:
29553:
29549:
29545:
29538:
29530:
29526:
29521:
29516:
29511:
29506:
29502:
29498:
29494:
29490:
29486:
29479:
29471:
29467:
29462:
29457:
29452:
29447:
29443:
29439:
29435:
29431:
29427:
29420:
29412:
29408:
29403:
29398:
29393:
29388:
29384:
29380:
29376:
29372:
29368:
29361:
29346:
29342:
29338:
29332:
29328:
29324:
29320:
29319:
29311:
29303:
29299:
29295:
29291:
29287:
29280:
29274:
29269:
29261:
29257:
29253:
29249:
29245:
29241:
29234:
29226:
29222:
29218:
29214:
29210:
29206:
29203:(10): 451–8.
29202:
29198:
29191:
29187:
29176:
29173:
29171:
29168:
29165:
29162:
29160:
29157:
29155:
29152:
29150:
29147:
29145:
29142:
29139:
29136:
29134:
29131:
29129:
29126:
29124:
29121:
29118:
29115:
29113:
29110:
29108:
29105:
29103:
29100:
29098:
29095:
29093:
29090:
29088:
29085:
29083:
29080:
29078:
29075:
29073:
29070:
29068:
29065:
29063:
29060:
29057:
29054:
29052:
29049:
29047:
29044:
29042:
29039:
29037:
29034:
29031:
29028:
29026:
29023:
29021:
29018:
29017:
29010:
29008:
29004:
29000:
28999:William Kahan
28996:
28992:
28988:
28984:
28980:
28975:
28973:
28969:
28952:
28948:
28939:
28935:
28931:
28926:
28924:
28920:
28916:
28915:Gale J. Young
28912:
28908:
28904:
28886:
28871:
28867:
28863:
28859:
28855:
28851:
28847:
28846:bilinear form
28843:
28809:
28767:
28764:
28763:
28762:
28760:
28755:
28740:
28706:
28702:
28664:
28660:
28628:
28597:
28593:
28568:
28565:
28554:
28541:
28536:
28532:
28522:
28516:
28512:
28503:
28500:
28496:
28490:
28486:
28480:
28476:
28472:
28467:
28463:
28453:
28447:
28443:
28434:
28431:
28426:
28414:
28410:
28400:
28394:
28390:
28386:
28383:
28378:
28366:
28362:
28353:
28350:
28337:
28322:
28319:
28295:
28266:
28262:
28229:
28225:
28195:
28191:
28159:
28102:
28078:
28068:
28058:
28057:is positive.
28040:
28028:
28024:
27992:
27969:
27954:
27942:
27938:
27929:
27924:
27909:
27896:
27894:
27889:
27886:
27871:
27863:
27859:
27849:
27845:
27838:
27828:
27810:
27806:
27779:
27775:
27738:
27734:
27716:
27706:
27673:
27661:
27657:
27640:
27625:
27619:
27616:
27613:
27605:
27601:
27590:
27575:
27547:
27543:
27531:
27516:
27504:
27500:
27491:
27478:
27462:
27438:
27432:
27429:
27426:
27399:
27370:
27356:
27338:
27309:
27306:
27271:
27251:
27231:
27210:
27194:
27165:
27110:
27071:
27055:
27039:
27010:
26955:
26916:
26864:
26862:
26861:Schatten norm
26857:
26844:
26837:
26825:
26822:
26818:
26807:
26804:
26800:
26776:
26755:
26752:
26748:
26741:
26712:
26697:
26693:
26689:
26688:
26669:
26638:
26634:
26622:
26605:
26600:
26596:
26590:
26586:
26567:
26548:
26545:
26532:
26526:
26515:
26504:
26500:
26497:
26492:
26479:
26466:
26447:
26444:
26431:
26427:
26416:
26405:
26401:
26398:
26395:
26384:
26368:
26352:
26349:
26346:
26336:
26326:
26303:
26272:
26268:
26264:
26249:
26236:
26233:
26230:
26211:
26206:
26191:
26162:
26157:
26153:
26149:
26134:
26100:
26096:
26092:
26077:
26059:
26058:normal matrix
26039:
26035:
26031:
26016:
25995:
25979:
25976:
25960:
25947:
25928:
25885:
25881:
25858:
25853:
25849:
25822:
25818:
25781:
25780:operator norm
25776:
25761:
25732:
25723:
25719:
25675:
25654:
25652:
25647:
25605:
25590:
25585:
25569:
25566:
25563:
25560:
25534:
25503:
25477:
25446:
25396:
25372:
25369:
25366:
25363:
25337:
25332:
25301:
25298:
25295:
25269:
25239:
25236:
25233:
25230:
25204:
25188:where matrix
25186:
25173:
25168:
25163:
25151:
25139:
25129:
25107:
25092:
25048:
25030:
25001:
24998:
24995:
24971:
24958:Truncated SVD
24955:
24940:
24937:
24934:
24931:
24905:
24900:
24869:
24866:
24863:
24837:
24807:
24804:
24801:
24798:
24772:
24741:
24735:
24732:
24729:
24720:
24717:
24691:
24632:
24599:
24568:
24518:
24507:
24494:
24489:
24484:
24472:
24460:
24450:
24437:
24403:
24387:
24372:
24354:
24348:
24345:
24342:
24333:
24330:
24319:
24304:
24301:
24298:
24295:
24269:
24264:
24233:
24230:
24227:
24201:
24171:
24168:
24165:
24162:
24136:
24105:
24076:
24050:
24019:
23964:
23938:
23905:
23889:the matrices
23876:
23870:
23867:
23864:
23855:
23852:
23843:
23830:
23825:
23820:
23808:
23796:
23786:
23773:
23739:
23723:
23673:
23670:
23667:
23651:
23645:
23639:
23633:
23627:
23621:
23615:
23606:
23583:
23573:
23568:
23564:
23558:
23554:
23550:
23547:
23541:
23536:
23526:
23521:
23517:
23511:
23507:
23503:
23500:
23494:
23489:
23479:
23474:
23470:
23464:
23460:
23456:
23453:
23447:
23442:
23432:
23427:
23423:
23417:
23413:
23409:
23406:
23400:
23395:
23385:
23380:
23376:
23370:
23366:
23362:
23359:
23353:
23348:
23338:
23333:
23329:
23323:
23319:
23315:
23312:
23304:
23301:
23296:
23284:
23280:
23271:
23266:
23254:
23250:
23241:
23236:
23224:
23220:
23211:
23206:
23194:
23190:
23179:
23177:
23163:
23151:
23146:
23142:
23138:
23133:
23128:
23124:
23120:
23115:
23110:
23106:
23102:
23097:
23092:
23088:
23079:
23074:
23060:
23048:
23044:
23035:
23030:
23018:
23014:
23005:
23000:
22988:
22984:
22975:
22970:
22958:
22954:
22936:
22931:
22919:
22915:
22906:
22901:
22889:
22885:
22876:
22871:
22859:
22855:
22846:
22841:
22829:
22825:
22814:
22812:
22805:
22801:
22788:
22786:
22768:
22764:
22712:
22707:
22703:
22693:
22678:
22674:
22668:
22664:
22660:
22655:
22651:
22645:
22641:
22637:
22632:
22628:
22622:
22618:
22614:
22604:
22600:
22596:
22569:
22566:
22563:
22547:
22544:
22540:
22488:
22483:
22434:
22429:
22373:
22330:
22326:
22322:
22318:
22314:
22309:
22296:
22291:
22276:
22247:
22237:
22221:
22211:
22181:
22176:
22127:
22125:
22121:
22117:
22099:
22096:
22093:
22083:
22065:
22062:
22059:
22049:
22048:GSL Team 2007
22045:
22041:
22036:
22034:
22030:
22026:
22022:
22017:
22015:
21995:
21991:
21987:
21984:
21979:
21975:
21971:
21968:
21958:
21914:
21890:
21866:
21862:
21856:
21852:
21848:
21845:
21840:
21836:
21832:
21829:
21819:
21814:
21812:
21789:
21785:
21781:
21775:
21748:
21742:
21715:
21709:
21699:
21695:
21691:
21673:
21670:
21667:
21664:
21635:
21631:
21627:
21621:
21612:
21609:. This takes
21608:
21558:
21515:
21498:
21446:
21422:
21379:
21360:
21338:
21335:
21332:are a set of
21305:
21304:
21303:
21295:
21281:
21278:
21272:
21252:
21247:
21243:
21239:
21233:
21213:
21193:
21184:
21170:
21150:
21141:
21128:
21125:
21122:
21117:
21113:
21106:
21098:
21096:
21092:
21076:
21068:
21064:
21054:
21040:
21020:
21012:
20996:
20976:
20954:
20950:
20946:
20943:
20940:
20937:
20928:
20914:
20891:
20888:
20885:
20882:
20879:
20876:
20873:
20870:
20867:
20864:
20861:
20858:
20855:
20829:
20826:
20823:
20800:
20780:
20757:
20754:
20751:
20748:
20745:
20739:
20719:
20710:
20697:
20691:
20688:
20685:
20682:
20679:
20673:
20670:
20664:
20656:
20654:
20639:
20636:
20620:
20615:
20582:
20491:
20480:
20469:
20466:
20449:
20444:
20432:
20428:
20424:
20422:
20415:
20389:
20384:
20372:
20368:
20364:
20362:
20355:
20331:
20328:
20315:
20310:
20306:
20302:
20299:
20294:
20290:
20286:
20283:
20278:
20274:
20265:
20251:
20248:
20234:
20192:
20152:
20134:
20117:
20112:
20100:
20096:
20092:
20089:
20086:
20083:
20081:
20074:
20048:
20045:
20042:
20037:
20025:
20021:
20017:
20014:
20012:
20005:
19981:
19978:
19945:
19940:
19936:
19932:
19909:
19904:
19900:
19896:
19868:
19865:
19854:
19852:
19848:
19833:
19828:
19824:
19771:
19738:
19722:
19718:
19700:
19667:
19628:
19617:
19591:
19587:
19562:
19557:
19524:
19491:
19487:
19462:
19457:
19454:
19451:
19447:
19394:
19391:
19388:
19384:
19333:
19323:
19319:
19299:
19296:
19293:
19289:
19262:
19259:
19256:
19252:
19227:
19222:
19219:
19216:
19212:
19208:
19203:
19200:
19197:
19193:
19168:
19157:
19144:
19139:
19136:
19133:
19129:
19125:
19101:
19096:
19093:
19090:
19086:
19082:
19058:
19033:
19022:
19011:
19004:, and define
18989:
18959:
18956:
18953:
18926:
18923:
18920:
18916:
18891:
18888:
18885:
18848:
18780:
18725:
18685:
18662:
18629:
18586:
18553:
18530:
18517:
18512:
18492:
18479:
18476:
18463:
18455:
18450:
18445:
18432:
18429:
18416:
18406:
18401:
18395:
18386:
18381:
18368:
18365:
18351:
18344:
18336:
18322:
18309:
18304:
18299:
18293:
18285:
18271:
18258:
18250:
18244:
18235:
18229:
18224:
18217:
18209:
18206:
18201:
18192:
18184:
18177:
18169:
18155:
18142:
18132:
18118:
18115:
18112:
18068:
18063:
18046:
18041:
18028:
18023:
18017:
18008:
18002:
17997:
17990:
17982:
17979:
17965:
17957:
17952:
17939:
17921:
17888:
17870:
17854:
17846:
17832:
17819:
17814:
17787:
17758:
17742:
17729:
17714:
17707:
17704:
17699:
17680:
17677:
17672:
17662:
17655:
17652:
17647:
17635:
17618:
17606:
17601:
17587:
17584:
17579:
17569:
17564:
17552:
17547:
17532:
17518:
17498:
17454:
17425:
17396:
17380:
17364:
17359:
17356:
17353:
17326:
17309:
17305:
17301:
17298:
17293:
17289:
17259:
17225:
17196:
17167:
17162:
17159:
17156:
17146:
17114:
17099:
17081:
17076:
17073:
17070:
17043:
17026:
17022:
17018:
17015:
17010:
17006:
16976:
16971:
16968:
16965:
16955:
16918:
16913:
16910:
16907:
16897:
16893:
16862:
16833:
16828:
16825:
16822:
16812:
16777:
16772:
16767:
16755:
16723:
16715:
16710:
16689:
16676:
16668:
16663:
16658:
16643:
16625:
16617:
16609:
16604:
16592:
16582:
16566:
16561:
16556:
16543:
16540:
16525:
16522:
16517:
16505:
16490:
16485:
16480:
16467:
16464:
16451:
16436:
16433:
16420:
16413:
16410:
16405:
16393:
16378:
16373:
16358:
16355:
16352:
16335:
16330:
16320:
16318:
16311:
16306:
16294:
16284:
16279:
16274:
16262:
16248:
16243:
16233:
16231:
16224:
16212:
16207:
16193:
16188:
16178:
16176:
16169:
16157:
16152:
16133:
16117:
16088:
16034:
16029:
16008:
15995:
15987:
15982:
15965:
15953:
15948:
15937:
15929:
15924:
15907:
15895:
15890:
15875:
15872:
15859:
15854:
15848:
15843:
15836:
15823:
15818:
15813:
15805:
15788:
15776:
15771:
15757:
15740:
15728:
15723:
15707:
15690:
15678:
15673:
15659:
15642:
15630:
15625:
15612:
15607:
15602:
15594:
15578:
15565:
15552:
15540:
15532:
15527:
15511:
15506:
15493:
15483:
15440:
15411:
15382:
15353:
15345:
15331:
15318:
15313:
15266:
15263:
15258:
15255:
15212:
15185:
15182:
15179:
15137:
15115:
15112:
15069:
15042:
15022:
14993:). Note that
14977:
14974:
14971:
14962:
14959:
14932:
14905:
14885:
14882:
14879:
14848:
14835:
14830:
14824:
14819:
14812:
14799:
14794:
14777:
14762:
14750:
14735:
14697:
14694:
14691:
14681:
14658:
14629:
14626:
14623:
14585:
14582:
14553:
14537:
14534:
14498:
14488:
14474:
14424:
14395:
14371:
14357:
14281:
14226:
14218:
14215:
14194:
14181:
14165:
14162:
14151:
14148:
14125:
14122:
14119:
14086:
14029:
13987:
13984:
13973:
13960:
13957:
13934:
13931:
13928:
13895:
13867:
13851:
13845:
13842:
13794:
13789:
13745:
13724:
13717:
13703:
13699:
13696:
13688:
13686:
13669:
13666:
13663:
13627:
13583:
13575:
13572:
13512:
13496:
13494:
13490:
13489:astrodynamics
13485:
13483:
13479:
13475:
13470:
13468:
13463:
13459:
13457:
13452:
13450:
13445:
13441:
13436:
13412:
13408:
13404:
13399:
13379:
13374:
13364:
13360:
13357:
13347:
13345:
13341:
13337:
13333:
13329:
13325:
13321:
13317:
13313:
13303:
13301:
13297:
13293:
13283:
13280:
13276:
13266:
13235:
13221:
13205:
13197:
13185:
13172:
13164:
13135:
13122:
13101:
13096:
13083:
13067:
13000:
12996:
12991:
12971:
12962:
12949:
12920:
12905:
12874:
12859:
12828:
12808:
12751:
12710:
12705:
12669:
12661:
12646:
12628:
12588:
12552:
12549:
12536:
12528:
12523:
12519:
12513:
12509:
12502:
12497:
12493:
12487:
12484:
12476:
12422:
12418:
12397:
12393:
12366:
12333:
12329:
12304:
12278:
12245:
12227:
12214:
12209:
12199:
12194:
12182:
12178:
12172:
12168:
12164:
12159:
12147:
12143:
12139:
12126:
12084:
12079:
12075:
12069:
12065:
12061:
12056:
12053:
12049:
12024:
12016:
12008:
11993:
11992:outer product
11957:
11955:
11951:
11950:
11931:
11867:
11854:
11849:
11820:
11798:
11782:
11755:
11752:
11749:
11718:
11715:
11646:
11628:
11618:
11615:, said to be
11557:
11556:approximating
11547:
11545:
11541:
11489:
11484:
11469:
11438:
11421:
11403:
11296:
11281:
11277:
11267:
11227:
11224:
11174:
11144:
11134:
11119:
11090:
11059:
11023:
11013:
10943:
10942:square matrix
10847:
10832:
10710:
10670:for a matrix
10650:
10630:
10620:
10618:
10614:
10574:
10558:
10545:
10540:
10528:
10513:
10508:
10493:
10475:
10455:
10414:
10413:pseudoinverse
10407:Pseudoinverse
10399:
10383:
10380:
10377:
10367:
10366:non-defective
10349:
10241:
10163:
10138:
10122:
10104:
10089:
10058:
10038:
10009:
9996:
9981:
9963:
9958:
9925:
9892:
9888:
9861:
9858:
9854:
9827:
9807:
9776:
9772:
9761:
9715:
9711:
9632:
9612:
9597:
9593:
9590:
9586:
9582:
9581:normal matrix
9535:
9530:
9487:
9470:
9440:
9424:
9419:
9371:
9355:
9345:
9328:
9298:
9297:
9296:
9293:
9276:
9271:
9256:
9233:
9228:
9216:
9198:
9178:
9176:
9169:
9150:
9145:
9125:
9107:
9102:
9079:
9067:
9052:
9050:
9038:
9019:
9003:
8998:
8978:
8936:
8934:
8930:
8912:
8909:
8906:
8890:
8875:
8662:
8659:
8655:
8643:
8575:
8551:
8548:
8545:
8536:is padded by
8495:
8492:
8489:
8486:
8462:
8459:
8456:
8406:
8403:
8400:
8376:
8373:
8370:
8367:
8357:
8354:which by the
8339:
8324:
8320:
8315:
8300:
8297:
8192:
8159:
8142:
8121:
8090:
8086:
8049:
8033:
8002:
7998:
7962:
7961:unitary basis
7958:
7941:
7891:
7888:
7885:
7875:
7874:
7873:
7802:
7799:
7796:
7787:
7784:
7760:
7718:
7703:
7683:
7670:
7667:
7651:
7648:
7638:
7634:
7633:left-singular
7577:
7560:
7552:
7549:
7547:
7535:
7521:
7513:
7510:
7508:
7486:
7468:
7464:
7411:
7407:
7344:
7343:
7324:
7303:
7300:
7285:
7276:
7265:
7256:
7249:
7240:
7235:
7224:
7219:
7210:
7201:
7194:
7185:
7172:
7163:
7156:
7149:
7140:
7129:
7122:
7115:
7108:
7101:
7092:
7085:
7078:
7071:
7064:
7057:
7052:
7047:
7032:
7003:
6984:
6974:
6969:
6963:
6958:
6953:
6948:
6943:
6936:
6931:
6926:
6921:
6916:
6909:
6904:
6899:
6894:
6889:
6882:
6877:
6872:
6867:
6862:
6855:
6850:
6845:
6840:
6835:
6829:
6824:
6822:
6815:
6794:
6784:
6779:
6773:
6768:
6763:
6758:
6751:
6746:
6741:
6736:
6729:
6724:
6719:
6714:
6707:
6702:
6697:
6692:
6686:
6681:
6679:
6672:
6648:
6646:
6626:
6583:
6579:
6559:
6489:
6470:
6461:
6452:
6445:
6438:
6429:
6424:
6415:
6408:
6401:
6394:
6387:
6378:
6371:
6364:
6357:
6354:
6347:
6336:
6331:
6324:
6317:
6310:
6301:
6296:
6287:
6280:
6273:
6270:
6263:
6256:
6249:
6244:
6242:
6235:
6219:
6193:
6188:
6183:
6165:
6160:
6155:
6150:
6132:
6127:
6120:
6113:
6095:
6090:
6085:
6080:
6074:
6069:
6067:
6051:
6044:
6037:
6034:
6027:
6020:
6011:
6008:
6001:
5994:
5987:
5978:
5971:
5964:
5957:
5954:
5945:
5938:
5931:
5928:
5921:
5914:
5909:
5907:
5888:
5871:
5843:
5828:
5822:
5817:
5812:
5807:
5802:
5795:
5790:
5785:
5780:
5775:
5768:
5763:
5758:
5753:
5748:
5741:
5736:
5731:
5726:
5721:
5715:
5710:
5697:
5681:
5678:
5675:
5666:Consider the
5659:
5644:
5641:
5615:
5605:
5597:
5568:
5562:
5556:
5506:
5500:
5494:
5468:
5458:
5426:
5420:
5385:
5367:
5362:
5329:
5298:
5272:
5238:
5232:
5208:
5184:
5181:
5178:
5175:
5165:
5147:
5142:
5125:
5107:
5083:
5078:
5047:
5036:
5020:
4996:
4991:
4987:
4962:
4936:
4932:
4907:
4883:
4857:
4853:
4826:
4822:
4795:
4791:
4782:
4778:
4774:
4771:
4760:
4745:
4739:
4736:
4733:
4724:
4721:
4697:
4694:
4686:
4673:
4649:
4620:
4594:
4590:
4578:
4565:
4559:
4556:
4553:
4544:
4541:
4538:
4535:
4532:
4529:
4525:
4520:
4508:
4504:
4500:
4492:
4479:
4471:
4468:
4448:
4438:
4433:
4424:
4420:
4414:
4407:
4403:
4395:
4391:
4388:
4380:
4379:
4374:
4372:
4352:
4348:
4295:
4285:
4282:
4279:
4274:
4241:
4237:
4226:
4180:
4170:
4167:
4164:
4159:
4015:
4012:
4009:
3999:
3970:
3950:
3893:
3883:
3864:
3847:
3803:
3800:
3797:
3787:
3758:
3714:
3704:
3703:
3697:
3695:
3691:
3647:
3580:
3577:
3547:
3527:
3496:
3468:
3463:
3433:
3404:
3373:
3342:
3337:
3325:
3321:
3294:
3252:. The matrix
3251:
3250:basis vectors
3247:
3227:
3170:
3168:
3150:
3140:
3137:-dimensional
3122:
3098:
3095:
3092:
3082:
3079:-dimensional
3064:
3054:
3053:square matrix
3037:
3034:
3031:
3021:
3018:-dimensional
3003:
2993:
2989:
2979:
2964:
2959:
2926:
2892:
2889:
2886:
2856:
2827:
2822:
2789:
2756:
2699:
2694:
2661:
2630:
2627:
2624:
2621:
2597:
2594:
2591:
2554:
2535:
2465:
2405:
2383:
2364:
2360:
2340:
2309:
2304:
2300:
2273:
2256:
2226:
2222:
2202:
2160:the matrices
2145:
2140:
2124:of the space
2087:
2069:
2054:
2036:
2033:
2030:
2004:
1961:
1960:square matrix
1943:
1940:
1937:
1891:
1869:
1864:
1860:
1852:
1843:
1837:
1811:
1806:
1802:
1797:
1792:
1787:
1782:
1778:
1773:
1769:
1764:
1755:
1753:
1749:
1748:least squares
1745:
1741:
1737:
1733:
1729:
1725:
1724:pseudoinverse
1720:
1707:
1702:
1692:
1682:
1672:
1662:
1647:
1630:
1627:
1624:
1592:
1575:
1572:
1569:
1519:
1487:
1484:
1481:
1472:
1469:
1445:
1442:
1439:
1436:
1384:
1368:
1353:
1348:
1333:
1230:
1227:
1213:
1198:
1168:
1165:
1162:
1153:
1150:
1141:
1128:
1123:
1118:
1106:
1094:
1090:
1084:
1079:
1076:
1073:
1069:
1065:
1052:
1036:
1032:
1009:
1004:
994:
991:
988:
983:
950:
940:
937:
934:
929:
913:
805:
775:
709:
706:
698:
693:
689:
679:
666:
634:
526:
508:
481:
478:
475:
441:
426:
423:
420:
390:
372:
369:
366:
318:
297:
246:
243:
240:
231:
229:
211:
208:
205:
195:
194:normal matrix
191:
187:
184:
180:
176:
175:factorization
172:
168:
164:
152:
147:
144:
137:
128:
122:
117:
114:
107:
98:
92:
86:
81:
78:
71:
62:
52:
47:
44:
43:
39:
29:
23:
19:
32275:Applications
32105:Disk algebra
31959:Spectral gap
31860:
31834:Main results
31682:Balanced set
31656:Distribution
31594:Applications
31447:Krein–Milman
31432:Closed graph
31106:Key concepts
31039:
31026:
30975:
30971:
30948:
30927:
30906:
30881:
30856:
30852:
30836:
30813:
30773:
30769:
30735:
30731:
30705:
30677:11311/959408
30659:
30655:
30632:
30614:
30571:
30567:
30560:Golub, G. H.
30554:
30541:
30511:(1): 51–90.
30508:
30504:
30495:
30473:(3): 211–8.
30470:
30464:
30455:
30444:the original
30435:
30425:
30380:
30376:
30366:
30341:11311/959408
30323:
30319:
30309:
30281:
30270:
30259:
30248:
30237:
30228:
30224:
30218:
30057:
30024:
30018:
30012:
29979:
29975:
29969:
29934:
29930:
29924:
29881:
29877:
29871:
29850:cite journal
29814:
29793:cite journal
29763:
29720:
29716:
29710:
29691:
29687:
29677:
29658:
29654:
29644:
29617:
29613:
29600:
29551:
29547:
29537:
29492:
29488:
29478:
29433:
29429:
29419:
29374:
29370:
29360:
29348:. Retrieved
29317:
29310:
29293:
29289:
29279:
29268:
29243:
29239:
29233:
29200:
29196:
29190:
28979:Kogbetliantz
28976:
28971:
28967:
28938:Émile Picard
28927:
28869:
28858:complete set
28839:
28765:
28756:
28555:
28338:
28064:
27970:
27897:
27890:
27887:
27829:
27641:
27532:
27479:
27216:
27056:
26875:
26858:
26777:
26695:
26691:
26685:
26623:
26549:
26546:
26448:
26445:
26369:
26332:
26207:
25996:
25929:
25777:
25665:
25662:Ky Fan norms
25648:
25586:
25187:
25108:
24961:
24508:
24438:
24436:is given by
24409:
24320:
23844:
23774:
23772:is given by
23745:
23656:
23649:
23643:
23637:
23631:
23625:
23619:
23609:Reduced SVDs
22789:
22694:
22553:
22313:QR algorithm
22310:
22238:
22128:
22037:
22021:QR algorithm
22018:
21815:
21578:
21337:eigenvectors
21301:
21185:
21142:
21099:
21060:
21009:is given by
20929:
20711:
20657:
20650:
20637:
20470:
20467:
20332:
20329:
20266:
20135:
19982:
19979:
19855:
19850:
19849:
19720:
19719:
19320:sets, their
19158:
18943:be the unit
18849:
18691:
18570:rather than
18531:
18480:
18477:
18133:
18044:
18042:
17940:
17871:
17869:is unitary.
17743:
17533:
17381:
17097:
16690:
16437:
16434:
16359:
16356:
16353:
16134:
16009:
15876:
15873:
15484:
14849:
14736:
14591:
14583:
14554:
14195:
14087:
13974:
13896:
13790:
13689:
13527:of a matrix
13502:
13486:
13471:
13464:
13460:
13453:
13437:
13400:
13348:
13309:
13289:
13272:
13222:
13186:
13084:
12992:
12558:
12550:
12477:
12417:Gabor filter
12228:
12127:
11963:
11954:Stewart 1993
11947:
11868:
11799:
11553:
11539:
11485:
11282:of a matrix
11273:
11175:of a vector
11140:
10626:
10559:
10494:
10410:
10123:
10121:is unitary.
9596:eigenvectors
9592:diagonalized
9552:
9327:eigenvectors
9294:
9020:
8937:
8896:
8644:
8316:
8140:
8138:
7932:has at most
7719:
7671:
7668:
7636:
7632:
7579:The vectors
7578:
7487:
7340:
7314:
7301:
7033:
7004:
6649:
6490:
5889:
5844:
5698:
5665:
5384:endomorphism
5037:
4761:
4579:
4472:
4469:
4381:
4375:
4090:
3757:column space
3184:
2985:
2555:
2466:
1901:
1867:
1850:
1841:
1835:
1804:
1795:
1785:
1771:
1721:
1648:, such that
1351:
1349:
1214:
1142:
1053:
680:
232:
192:of a square
170:
166:
160:
150:
145:
135:
126:
120:
115:
105:
96:
90:
84:
79:
69:
60:
50:
45:
37:
27:
18:
32302:Heat kernel
32002:Compression
31887:Isospectral
31611:Heat kernel
31601:Hardy space
31508:Trace class
31422:Hahn–Banach
31384:Topological
30972:SIAM Review
30902:"Chapter 3"
30564:Reinsch, C.
29159:Time series
28911:Carl Eckart
28831:is compact.
25718:matrix norm
24756:The matrix
24406:Compact SVD
24120:The matrix
23979:columns of
22783:denote the
22046:in step 2 (
21497:eigenvalues
21334:orthonormal
21011:normalising
20216:and taking
19242:Since both
18975:-sphere in
17098:orthonormal
13340:mode shapes
12614:closest to
10997:satisfying
10862:The vector
10803:belongs to
10696:and vector
10364:While only
9469:eigenvalues
7820:columns of
7631:are called
4030:columns of
3908:columns of
3818:columns of
3729:columns of
2359:composition
1736:engineering
1352:compact SVD
142:vertically.
32387:Categories
31980:Continuous
31795:C*-algebra
31790:B*-algebra
31544:C*-algebra
31359:Properties
31148:algorithms
30607:References
30461:Eckart, C.
30253:Netlib.org
30242:Netlib.org
29730:1810.07060
29614:Math. Comp
29350:19 January
28995:Gene Golub
28862:invariants
28286:. For any
27477:such that
27385:a unitary
26210:trace norm
19721:Statement.
18876:denote an
18045:or removed
17802:such that
14734:such that
14196:Therefore
14085:we obtain
13654:be a real
13143:subject to
11280:null space
10831:null space
10619:problems.
10613:reciprocal
9982:theorem:
8141:degenerate
7775:The first
7485:such that
7031:such that
4874:such that
3884:The first
3846:null space
3705:The first
2225:reflection
2219:represent
2053:orthogonal
1742:, such as
1740:statistics
1732:null space
633:orthogonal
31:of a real
31766:-algebras
31518:Unbounded
31513:Transpose
31471:Operators
31400:Separable
31395:Reflexive
31380:Algebraic
31366:Barrelled
31175:CPU cache
30980:CiteSeerX
30740:CiteSeerX
30596:123532178
30192:⇔
30183:‖
30177:‖
30151:‖
30130:‖
30102:∗
30090:∗
30072:
30049:239490016
30004:252637213
29944:1406.3496
29891:1406.3506
29833:1304.1467
29755:118935941
29592:Highlight
28949:σ
28928:In 1907,
28810:∗
28594:σ
28501:ψ
28487:σ
28477:∑
28432:ψ
28427:∗
28391:∑
28384:ψ
28379:∗
28351:ψ
28296:ψ
28263:σ
28160:∗
28103:λ
28041:∗
27993:∗
27955:∗
27930:⋅
27925:∗
27707:∗
27674:∗
27620:μ
27517:∗
27433:μ
27252:∗
27241:Σ
26801:∑
26635:σ
26597:σ
26587:∑
26565:‖
26557:‖
26516:∗
26501:
26488:⟩
26472:⟨
26464:‖
26456:‖
26417:∗
26402:
26393:⟩
26377:⟨
26350:×
26304:∗
26250:∗
26237:
26228:‖
26220:‖
26135:∗
26089:‖
26078:∗
26068:‖
26017:∗
25972:‖
25961:∗
25951:‖
25945:‖
25937:‖
25882:ℓ
25747:-norm of
25564:≪
25530:Σ
25478:∗
25387:Only the
25367:×
25338:∗
25299:×
25265:Σ
25234:×
25169:∗
25147:Σ
25124:~
25066:~
24999:≪
24935:×
24906:∗
24867:×
24833:Σ
24802:×
24721:≪
24692:∗
24628:Σ
24600:∗
24509:Only the
24490:∗
24468:Σ
24334:≪
24299:×
24270:∗
24231:×
24197:Σ
24166:×
24102:Σ
24046:Σ
23826:∗
23804:Σ
23671:×
23574:∗
23551:
23527:∗
23504:
23480:∗
23457:
23433:∗
23410:
23386:∗
23363:
23339:∗
23316:
23302:±
23139:−
23121:−
23103:−
23080:−
22937:±
22806:±
22802:σ
22765:σ
22713:∈
22675:σ
22652:σ
22629:σ
22567:×
22518:⇐
22484:∗
22464:⇒
22430:∗
22415:⇒
22346:⇒
22277:∗
22212:∗
22177:∗
22097:×
22063:×
21846:−
21692:(as with
21668:≥
21559:∗
21516:∗
21482:Σ
21423:∗
21361:∗
21276:←
21237:←
21110:←
20889:…
20865:…
20758:θ
20720:θ
20692:θ
20668:←
20481:σ
20429:σ
20369:σ
20307:λ
20291:λ
20275:σ
20246:‖
20238:‖
20232:‖
20224:‖
20097:λ
20022:λ
19949:∇
19946:⋅
19937:λ
19933:−
19913:∇
19910:⋅
19901:λ
19897:−
19872:∇
19866:σ
19863:∇
19825:σ
19618:σ
19588:σ
19488:σ
19455:−
19392:−
19334:σ
19297:−
19260:−
19220:−
19209:×
19201:−
19169:σ
19137:−
19126:∈
19094:−
19083:∈
19012:σ
18957:−
18924:−
18889:×
18663:∗
18630:∗
18587:∗
18554:∗
18513:∗
18502:Σ
18451:∗
18387:∗
18300:∗
18116:×
18105:equal to
18092:Σ
17949:Σ
17700:−
17673:−
17648:−
17619:∗
17607:∗
17580:−
17553:∗
17365:ℓ
17299:−
17290:λ
17226:∗
17168:ℓ
17082:ℓ
17016:−
17007:λ
16977:ℓ
16919:ℓ
16894:λ
16863:∗
16834:ℓ
16773:∗
16664:∗
16626:−
16610:∗
16583:−
16562:∗
16518:−
16486:∗
16406:−
16312:∗
16280:∗
16213:∗
16158:∗
15966:∗
15954:∗
15908:∗
15896:∗
15789:∗
15777:∗
15741:∗
15729:∗
15691:∗
15679:∗
15643:∗
15631:∗
15553:∗
15533:∗
15512:∗
15441:∗
15249:¯
15213:∗
15186:ℓ
15106:¯
15070:∗
14963:≤
14960:ℓ
14933:∗
14906:ℓ
14886:ℓ
14883:×
14880:ℓ
14789:¯
14763:∗
14751:∗
14695:×
14659:∗
14627:×
14499:∗
14396:λ
14219:λ
14166:λ
14163:−
14129:∇
14126:⋅
14123:λ
14120:−
14095:∇
14026:, is the
14012:∇
13985:λ
13938:∇
13935:⋅
13932:λ
13929:−
13904:∇
13868:theorem,
13840:‖
13832:‖
13748:↦
13727:→
13687:. Define
13667:×
13626:Hermitian
13576:λ
13513:λ
13422:Σ
13411:entangled
13380:σ
13365:σ
13358:κ
13326:, and in
13202:‖
13198:⋅
13195:‖
13161:Ω
13150:Ω
13132:‖
13123:−
13119:Ω
13110:‖
13105:Ω
12997:, is the
12921:∗
12875:∗
12829:∗
12818:Σ
12752:∗
12706:∗
12662:−
12520:σ
12510:∑
12494:σ
12485:α
12394:σ
12330:σ
12200:⊗
12179:σ
12169:∑
12144:∑
12017:⊗
11909:Σ
11885:~
11881:Σ
11850:∗
11837:~
11833:Σ
11815:~
11737:~
11719:
11691:~
11617:truncated
11600:~
11558:a matrix
11525:Σ
11439:∗
11222:‖
11214:‖
11060:∗
11014:∗
10627:A set of
10598:Σ
10570:Σ
10541:∗
10524:Σ
10476:∗
10465:Σ
10381:×
10268:Σ
10105:∗
10059:∗
10048:Σ
9889:σ
9862:φ
9828:∗
9773:σ
9712:σ
9633:∗
9589:unitarily
9531:∗
9488:∗
9452:Σ
9420:∗
9346:∗
9272:∗
9257:∗
9252:Σ
9246:Σ
9229:∗
9217:∗
9212:Σ
9199:∗
9188:Σ
9170:∗
9146:∗
9132:Σ
9126:∗
9121:Σ
9103:∗
9092:Σ
9080:∗
9068:∗
9063:Σ
9039:∗
8999:∗
8988:Σ
8910:×
8663:φ
8549:−
8460:−
8371:≠
8298:σ
8273:Σ
7889:×
7731:Σ
7704:∗
7693:Σ
7649:σ
7553:σ
7536:∗
7514:σ
7325:σ
7236:−
7220:−
7048:∗
6816:∗
6673:∗
6627:∗
6560:∗
6502:Σ
6425:−
6355:−
6332:−
6297:−
6271:−
6236:∗
6062:Σ
6035:−
6009:−
5955:−
5929:−
5872:∗
5861:Σ
5679:×
5616:∗
5606:∘
5598:∘
5469:∗
5459:∘
5330:∗
5164:semi-axes
5124:ellipsoid
4898:maps the
4788:→
4646:Σ
4591:σ
4542:…
4505:σ
4441:↦
4417:→
4283:…
4223:yield an
4168:…
4013:−
4000:The last
3969:row space
3951:∗
3865:∗
3801:−
3788:The last
3528:∗
3464:∗
3374:∗
3322:σ
3228:∗
3139:ellipsoid
3096:×
3081:ellipsoid
3035:×
2853:Σ
2757:∗
2625:≠
2595:×
2536:∗
2418:Σ
2384:∗
2301:σ
2238:Σ
2221:rotations
2203:∗
2102:↦
2034:×
2005:∗
1941:×
1871:, namely
1859:semi-axes
1820:Σ
1777:unit disc
1683:∗
1663:∗
1628:×
1573:×
1473:≤
1440:×
1412:Σ
1401:in which
1385:∗
1377:Σ
1267:(but not
1254:Σ
1224:Σ
1154:≤
1124:∗
1091:σ
1070:∑
1033:σ
992:…
938:…
733:Σ
703:Σ
690:σ
649:Σ
509:∗
479:×
424:×
400:Σ
370:×
313:∗
305:Σ
244:×
209:×
32367:Weyl law
32312:Lax pair
32259:Examples
32093:With an
32012:Discrete
31990:Residual
31926:Spectrum
31911:operator
31903:operator
31895:operator
31810:Spectrum
31720:Category
31532:Algebras
31414:Theorems
31371:Complete
31340:Schwartz
31286:glossary
31204:Software
31168:Hardware
31127:Problems
31021:(2003).
31002:1903/566
30873:37591557
30812:(1996).
30694:14714823
30686:26357324
30487:10163399
30433:(2018),
30417:16578416
30358:14714823
30350:26357324
29961:17966555
29916:15476557
29776:(2000).
29588:24282503
29548:PLOS ONE
29529:16877539
29470:15545604
29411:10963673
29345:52286352
29296:: 1–11.
29260:11247991
29225:12827601
29013:See also
28983:Hestenes
28766:Theorem.
28523:⟩
28497:⟨
28454:⟩
28401:⟨
24789:is thus
24153:is thus
23742:Thin SVD
23714:of rank
22370:and the
22319:and the
21499:of both
20907:, where
13474:outbreak
13442:, where
13316:Tikhonov
13300:big data
13277:(called
12972:′
12295:are the
9878:of each
9579:being a
8964:has SVD
8319:cokernel
5289:sent by
4091:Because
1801:rotation
387:complex
31908:Unitary
31523:Unitary
31503:Nuclear
31488:Compact
31483:Bounded
31478:Adjoint
31452:Min–max
31345:Sobolev
31330:Nuclear
31320:Hilbert
31315:Fréchet
31280: (
31010:2132388
30798:2949777
30778:Bibcode
30588:1553974
30533:0092215
30525:2098862
30408:1063464
30385:Bibcode
30029:Bibcode
29984:Bibcode
29896:Bibcode
29838:Bibcode
29735:Bibcode
29622:Bibcode
29579:3839928
29556:Bibcode
29520:1524674
29497:Bibcode
29438:Bibcode
29379:Bibcode
29217:8545912
29166:(2DSVD)
28903:Autonne
28899:
28874:
28836:History
28829:
28795:
28791:
28769:
28753:
28728:
28724:
28686:
28684:(resp.
28682:
28644:
28641:
28616:
28612:
28585:
28581:
28558:
28335:
28312:
28308:
28288:
28284:
28251:
28247:
28214:
28210:
28183:
28179:
28145:
28141:
28119:
28115:
28095:
28091:
28071:
28055:
28011:
28007:
27973:
27825:
27798:
27794:
27767:
27763:
27741:
27729:
27692:
27688:
27644:
27638:
27593:
27588:
27568:
27564:is the
27562:
27535:
27475:
27455:
27451:
27416:
27412:
27387:
27383:
27358:
27351:
27326:
27322:
27299:
27295:
27273:
27266:
27219:
27207:
27182:
27178:
27153:
27149:
27127:
27123:
27088:
27084:
27059:
27052:
27027:
27023:
26998:
26994:
26972:
26968:
26933:
26929:
26904:
26900:
26878:
26773:
26729:
26725:
26700:
26682:
26657:
26653:
26626:
26365:
26339:
26323:
26289:
26204:
26179:
26175:
26117:
26054:
25999:
25925:
25903:
25871:
25841:
25837:
25810:
25806:
25784:
25774:
25749:
25745:
25725:
25714:
25692:
25688:
25668:
25644:
25622:
25618:
25593:
25582:
25553:
25549:
25520:
25516:
25496:
25492:
25463:
25459:
25439:
25435:
25413:
25409:
25389:
25385:
25356:
25352:
25318:
25314:
25288:
25284:
25255:
25252:
25223:
25219:
25190:
25105:
25085:
25081:
25050:
25043:
25018:
25014:
24988:
24984:
24964:
24953:
24924:
24920:
24886:
24882:
24856:
24852:
24823:
24820:
24791:
24787:
24758:
24754:
24710:
24706:
24677:
24673:
24651:
24647:
24618:
24614:
24585:
24581:
24561:
24557:
24535:
24531:
24511:
24434:
24412:
24400:
24375:
24367:
24323:
24317:
24288:
24284:
24250:
24246:
24220:
24216:
24187:
24184:
24155:
24151:
24122:
24118:
24093:
24089:
24069:
24065:
24036:
24032:
24007:
24003:
23981:
23977:
23957:
23953:
23924:
23920:
23891:
23770:
23748:
23736:
23716:
23712:
23690:
23688:matrix
23686:
23660:
22754:
22732:
22582:
22556:
22535:
22505:
22503:update
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5694:
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2404:scaling
2398:
2369:
2357:into a
2355:
2326:
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2259:
2255:scaling
2251:
2229:
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2057:
2049:
2023:
2019:
1990:
1986:
1964:
1956:
1930:
1926:
1904:
1857:of the
1810:scaling
1791:ellipse
1643:
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577:
555:
551:
529:
523:is the
466:
444:
385:
359:
355:
333:
283:
261:
224:
198:
183:complex
173:) is a
116:Bottom:
35:matrix
31892:Normal
31498:Normal
31335:Orlicz
31325:Hölder
31305:Banach
31294:Spaces
31282:topics
31226:LAPACK
31216:MATLAB
31046:
31008:
30982:
30955:
30936:
30913:
30888:
30871:
30820:
30796:
30742:
30712:
30692:
30684:
30639:
30621:
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30586:
30531:
30523:
30485:
30415:
30405:
30356:
30348:
30297:
30231:: 359.
30047:
30002:
29959:
29914:
29772:&
29753:
29586:
29576:
29527:
29517:
29468:
29461:534520
29458:
29409:
29399:
29343:
29333:
29258:
29223:
29215:
29119:(MPCA)
29058:(EOFs)
27533:where
26624:where
25722:Ky Fan
25720:, the
23845:where
22695:where
22400:gives
22331:gives
22327:. The
22029:LAPACK
22027:. The
21143:where
19851:Proof.
19577:Since
16691:since
16435:Then,
15172:, for
14850:where
14614:be an
13187:where
13102:argmin
12768:where
11869:where
11173:2-norm
10560:where
10257:where
10179:where
10024:where
8323:kernel
8143:. If
4580:where
3692:, its
3185:Since
1928:is an
1738:, and
1730:, and
1615:is an
1560:is an
1460:where
1143:where
468:is an
413:is an
357:is an
331:where
186:matrix
165:, the
146:Right:
103:, and
31985:Point
31310:Besov
31211:ATLAS
31006:JSTOR
30869:S2CID
30794:JSTOR
30690:S2CID
30592:S2CID
30521:JSTOR
30483:S2CID
30447:(PDF)
30440:(PDF)
30354:S2CID
30045:S2CID
30000:S2CID
29957:S2CID
29939:arXiv
29912:S2CID
29886:arXiv
29828:arXiv
29824:(PDF)
29781:(PDF)
29751:S2CID
29725:arXiv
29610:(PDF)
29402:27718
29341:S2CID
29221:S2CID
29182:Notes
29140:(PCA)
29077:K-SVD
26694:, or
26060:, so
26056:is a
25716:is a
25657:Norms
21611:order
13598:When
12229:Here
11276:range
11043:with
10940:is a
8510:then
7339:is a
3967:(the
3574:is a
3167:below
2943:into
2714:Then
2612:with
2084:as a
1958:real
1728:range
177:of a
80:Left:
31916:Unit
31764:and
31658:(or
31376:Dual
31190:SIMD
31044:ISBN
30953:ISBN
30934:ISBN
30911:ISBN
30886:ISBN
30818:ISBN
30710:ISBN
30682:PMID
30637:ISBN
30619:ISBN
30413:PMID
30346:PMID
30295:ISBN
29863:help
29806:help
29584:PMID
29525:PMID
29489:PNAS
29466:PMID
29430:PNAS
29407:PMID
29371:PNAS
29352:2023
29331:ISBN
29256:PMID
29213:PMID
29032:(CA)
28997:and
28913:and
28852:and
27735:for
27151:and
26996:and
25839:and
25437:and
24675:and
24559:and
24034:and
24005:and
23922:and
23647:and
23635:and
22541:and
22236:or
21537:and
21206:and
20793:and
20599:and
20540:and
19755:and
19541:and
19411:and
19316:are
19279:and
18850:Let
18821:and
18766:and
18646:and
17872:For
17511:and
17411:and
16103:and
15397:and
15183:>
14592:Let
13330:and
13298:and
13273:The
12262:and
11716:rank
11673:and
11488:rank
11278:and
10492:is,
10309:and
8835:and
8783:and
8490:<
8404:>
8321:and
8236:and
8176:and
8076:for
7988:for
7846:and
7639:for
7635:and
7605:and
7428:and
7345:for
6610:and
6576:are
6543:and
4843:and
4725:>
4712:for
4664:and
4376:The
4117:and
3607:and
3544:are
3511:and
3390:and
3211:and
3177:and
2806:and
2740:and
2678:to
2519:and
2186:and
1988:and
1878:and
1848:and
1808:, a
1593:and
1293:and
967:and
804:rank
605:and
179:real
67:and
46:Top:
31180:TLB
30998:hdl
30990:doi
30861:doi
30853:BIT
30786:doi
30750:doi
30672:hdl
30664:doi
30576:doi
30513:doi
30475:doi
30403:PMC
30393:doi
30336:hdl
30328:doi
30287:doi
30037:doi
29992:doi
29949:doi
29904:doi
29743:doi
29696:doi
29663:doi
29659:326
29630:doi
29574:PMC
29564:doi
29515:PMC
29505:doi
29493:103
29456:PMC
29446:doi
29434:101
29397:PMC
29387:doi
29323:doi
29298:doi
29248:doi
29205:doi
28974:).
28860:of
28310:in
27591:on
26698:of
25782:of
25354:is
25286:is
25221:is
24922:is
24854:is
24724:min
24373:of
24337:max
24286:is
24218:is
23859:min
22374:of
21339:of
19684:or
19437:in
19374:in
15301:as
14966:min
14386:so
14322:of
14241:so
14028:del
13624:is
13487:In
13384:min
13369:max
13053:to
12647:of
11956:).
11490:of
10829:'s
9942:or
9758:is
9509:or
9471:of
9329:of
8938:If
8616:or
7876:An
7791:min
7454:in
7397:in
7277:0.1
7266:0.5
7241:0.4
7225:0.1
7211:0.5
7186:0.4
7173:0.8
7141:0.2
6462:0.2
6430:0.8
6337:0.8
6302:0.2
5126:in
4728:min
4548:min
4312:of
4227:of
4197:of
3971:of
3848:of
3759:of
3141:in
2881:min
2223:or
1882:2,2
1875:1,1
1865:of
1476:min
1157:min
806:of
776:of
724:of
527:of
181:or
171:SVD
161:In
28:UΣV
32389::
31284:–
31038:,
31004:.
30996:.
30988:.
30976:35
30974:.
30970:.
30905:.
30867:.
30857:27
30855:.
30835:.
30808:;
30792:.
30784:.
30772:.
30764:;
30748:.
30736:11
30734:.
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30688:.
30680:.
30670:.
30660:12
30658:.
30654:.
30590:.
30584:MR
30582:.
30572:14
30570:.
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30529:MR
30527:.
30519:.
30507:.
30481:.
30469:.
30411:.
30401:.
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30379:.
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30344:.
30334:.
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30322:.
30318:.
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30279:.
30229:10
30227:.
30069:Tr
30043:.
30035:.
30025:69
30023:.
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29990:.
29978:.
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29947:.
29935:19
29933:.
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29902:.
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29882:32
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29373:.
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29242:.
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29211:.
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29199:.
28925:.
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26498:tr
26399:tr
26234:Tr
25653:.
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23548:Im
23501:Im
23454:Im
23407:Re
23360:Re
23313:Re
21294:.
21265:,
21053:.
20655:,
17379:.
16331:12
14057:).
13495:.
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13294:,
13265:.
11228:1.
11141:A
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6647:.
3581:,
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2433:),
2400:),
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30360:.
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30289::
30213:.
30201:0
30198:=
30195:A
30189:0
30186:=
30180:A
30155:2
30145:2
30140:V
30134:M
30127:=
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30119:2
30114:V
30108:M
30097:M
30085:2
30080:V
30075:(
30051:.
30039::
30031::
30006:.
29994::
29986::
29980:7
29963:.
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29906::
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29531:.
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29413:.
29389::
29381::
29354:.
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1805:V
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121:Σ
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41:.
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