10428:
9772:
10423:{\displaystyle {\begin{aligned}\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\\\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\end{aligned}}}
9734:
9139:
5927:
10775:
9333:
8739:
15932:
17038:
18247:
5628:
6844:
10483:
9729:{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&-\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}\\-\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&\quad \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}\end{bmatrix}}.}
15585:
11299:
9134:{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}&-\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\\-\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}&\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\end{bmatrix}},}
7283:
15596:
5922:{\displaystyle \mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\mathbf {C} ^{\mathrm {T} }={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}{\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{21}&\cdots &\mathbf {C} _{n1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1n}&\mathbf {C} _{2n}&\cdots &\mathbf {C} _{nn}\\\end{pmatrix}}}
17302:
10770:{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {A} -\mathbf {B} \mathbf {D} ^{-1}\mathbf {C} \right)^{-1}&\mathbf {0} \\\mathbf {0} &\left(\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {B} \right)^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {I} &-\mathbf {B} \mathbf {D} ^{-1}\\-\mathbf {C} \mathbf {A} ^{-1}&\mathbf {I} \end{bmatrix}}.}
6536:
15291:
11018:
77:
6871:
15927:{\displaystyle {\begin{aligned}(\mathbf {A} +\varepsilon \mathbf {X} )^{n}&=\mathbf {A} ^{n}+\varepsilon \sum _{i=1}^{n}\mathbf {A} ^{i-1}\mathbf {X} \mathbf {A} ^{n-i}+{\mathcal {O}}\left(\varepsilon ^{2}\right),\\(\mathbf {A} +\varepsilon \mathbf {X} )^{-n}&=\mathbf {A} ^{-n}-\varepsilon \sum _{i=1}^{n}\mathbf {A} ^{-i}\mathbf {X} \mathbf {A} ^{-(n+1-i)}+{\mathcal {O}}\left(\varepsilon ^{2}\right).\end{aligned}}}
14577:
36:
7950:
6389:
16771:
179:
4608:
13546:
10999:
6839:{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,I\\\end{bmatrix}}^{\mathrm {T} }={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,I\\\end{bmatrix}}}
6100:
15580:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {A} ^{n}}{\mathrm {d} t}}&=\sum _{i=1}^{n}\mathbf {A} ^{i-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{n-i},\\{\frac {\mathrm {d} \mathbf {A} ^{-n}}{\mathrm {d} t}}&=-\sum _{i=1}^{n}\mathbf {A} ^{-i}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-(n+1-i)}.\end{aligned}}}
11294:{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {C} ^{T}\\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&-\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\\-\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&\mathbf {S} ^{-1}\end{bmatrix}},}
7278:{\displaystyle {\begin{alignedat}{6}A&={}&(ei-fh),&\quad &D&={}&-(bi-ch),&\quad &G&={}&(bf-ce),\\B&={}&-(di-fg),&\quad &E&={}&(ai-cg),&\quad &H&={}&-(af-cd),\\C&={}&(dh-eg),&\quad &F&={}&-(ah-bg),&\quad &I&={}&(ae-bd).\\\end{alignedat}}}
14390:
8717:
12905:
7738:
12573:
1793:
15260:
6167:
14959:
4321:
7547:
15101:
13335:
10832:
5067:
15993:
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of
5938:
8382:
14791:
14326:
11767:
12368:
14572:{\displaystyle {\frac {\mathrm {d} (\mathbf {A} ^{-1}\mathbf {A} )}{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}\mathbf {A} +\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {I} }{\mathrm {d} t}}=\mathbf {0} .}
12713:
8411:
4279:; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to
12738:
3983:
3298:
7945:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}{(\mathbf {x} _{1}\times \mathbf {x} _{2})}^{\mathrm {T} }\\{(\mathbf {x} _{2}\times \mathbf {x} _{0})}^{\mathrm {T} }\\{(\mathbf {x} _{0}\times \mathbf {x} _{1})}^{\mathrm {T} }\end{bmatrix}}.}
12407:
3460:
6384:{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{\det \mathbf {A} }}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}.}
3083:
1613:
7643:
15112:
4274:
above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining
3841:
11639:
11540:
14822:
4603:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(\mathbf {A} ^{l}\right)^{k_{l}},}
2855:
13315:
3640:
4126:
2249:
8086:
8232:
7371:
14159:
13541:{\displaystyle \mathbf {x} ^{i}=x_{ji}\mathbf {e} ^{j}=(-1)^{i-1}(\mathbf {x} _{1}\wedge \cdots \wedge ()_{i}\wedge \cdots \wedge \mathbf {x} _{n})\cdot (\mathbf {x} _{1}\wedge \ \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})^{-1}}
11929:
10994:{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {0} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}&\mathbf {0} \\-\mathbf {D} ^{-1}\mathbf {CA} ^{-1}&\mathbf {D} ^{-1}\end{bmatrix}}.}
14970:
2990:
2695:
4851:
6095:{\displaystyle \left(\mathbf {A} ^{-1}\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} ^{\mathrm {T} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ji}\right)}
5176:
12196:
11380:
5551:
8256:
1451:
14678:
14213:
14638:
3716:
14053:
11643:
8091:
The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of
2597:
12215:
13812:
1550:
6513:
12095:
1353:
4843:
8712:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left({\frac {1}{6}}\left\mathbf {I} -{\frac {1}{2}}\mathbf {A} \left+\mathbf {A} ^{2}\operatorname {tr} \mathbf {A} -\mathbf {A} ^{3}\right).}
4041:
1859:
11447:
12900:{\displaystyle \mathbf {A} ^{-1}=\mathbf {X} ^{-1}-{\frac {\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\mathbf {X} ^{-1}}{1+\operatorname {tr} \left(\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\right)}}~.}
12590:
5361:
348:
1280:
3883:
3206:
14378:
8147:
2082:
1600:
13154:
6876:
5441:
1207:
12568:{\displaystyle \lim _{n\to \infty }\left(\mathbf {I} -\mathbf {X} ^{-1}\mathbf {A} \right)^{n}=0\mathrm {~~or~~} \lim _{n\to \infty }\left(\mathbf {I} -\mathbf {A} \mathbf {X} ^{-1}\right)^{n}=0}
15601:
15296:
9777:
1114:
1788:{\displaystyle (\mathbf {A} _{1}\mathbf {A} _{2}\cdots \mathbf {A} _{k-1}\mathbf {A} _{k})^{-1}=\mathbf {A} _{k}^{-1}\mathbf {A} _{k-1}^{-1}\cdots \mathbf {A} _{2}^{-1}\mathbf {A} _{1}^{-1}.}
3371:
15255:{\displaystyle f(\mathbf {A} +\varepsilon \mathbf {X} )=f(\mathbf {A} )+\varepsilon \sum _{i}g_{i}(\mathbf {A} )\mathbf {X} h_{i}(\mathbf {A} )+{\mathcal {O}}\left(\varepsilon ^{2}\right).}
4761:
2997:
2761:
16793:
11966:
11823:
7567:
7360:
2358:
2300:
5293:
4251:
3756:
3366:
14954:{\displaystyle \left(\mathbf {A} +\varepsilon \mathbf {X} \right)^{-1}=\mathbf {A} ^{-1}-\varepsilon \mathbf {A} ^{-1}\mathbf {X} \mathbf {A} ^{-1}+{\mathcal {O}}(\varepsilon ^{2})\,.}
11544:
2418:
11451:
3201:
3564:
1938:
13606:
13051:
3878:
14670:
3511:
2772:
643:, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a
16279:
14082:
13694:
13665:
13223:
13187:
7730:
7701:
7672:
3751:
2012:
13844:
13228:
13089:
647:, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
3569:
4049:
803:
is invertible, i.e., has an inverse under function composition. (Here, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible")
14814:
14181:
11845:
11789:
7542:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left({\frac {1}{2}}\left\mathbf {I} -\mathbf {A} \operatorname {tr} \mathbf {A} +\mathbf {A} ^{2}\right).}
2880:
2720:
750:
4680:
2176:
16856:
8006:
3720:
The reason it works is that the process of
Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using
8155:
1075:
15096:{\displaystyle {\frac {\mathrm {d} f(\mathbf {A} )}{\mathrm {d} t}}=\sum _{i}g_{i}(\mathbf {A} ){\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}h_{i}(\mathbf {A} ),}
14087:
11850:
13636:
5231:
3139:
3112:
1971:
5062:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=1}^{n}\mathbf {A} ^{s-1}{\frac {(-1)^{n-1}}{(n-s)!}}B_{n-s}(t_{1},t_{2},\ldots ,t_{n-s}).}
2916:
2621:
15283:
2910:. Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.
8377:{\displaystyle 1={\frac {1}{\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}}\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}
5108:
12116:
11320:
14786:{\displaystyle {\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.}
14321:{\displaystyle {\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.}
5473:
1376:
788:. (In this statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible", in which one-sided inverses are considered.)
14585:
3647:
13853:
11762:{\displaystyle \mathbf {W} _{4}=\mathbf {W} _{1}^{T}\mathbf {W} _{3}=\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},}
17905:
11986:
12363:{\displaystyle \sum _{n=0}^{2^{L}-1}(\mathbf {I} -\mathbf {A} )^{n}=\prod _{l=0}^{L-1}\left(\mathbf {I} +(\mathbf {I} -\mathbf {A} )^{2^{l}}\right)}
10826:
in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes
299:
197:
2538:
13699:
1478:
16634:"Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems"
6419:
12035:
9243:
1290:
16466:. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002.
5073:
4773:
440:
is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the
12708:{\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }\left(\mathbf {X} ^{-1}(\mathbf {X} -\mathbf {A} )\right)^{n}\mathbf {X} ^{-1}~.}
6865:
If the determinant is non-zero, the matrix is invertible, with the entries of the intermediary matrix on the right side above given by
16835:
3988:
1799:
11397:
5314:
3978:{\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} =\mathbf {I} \mathbf {A} ^{-1}.}
3293:{\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).}
18119:
17338:
16896:
1213:
18210:
17229:
14338:
8107:
141:
17287:
3578:
3380:
3215:
3006:
2513:, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be
2021:
1555:
113:
16616:
16493:
16432:
16407:
16351:
16239:
16170:
13102:
754:
of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:
15985:
are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.
18129:
17895:
5375:
4144:. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes
1156:
16261:
120:
3455:{\displaystyle \left({\begin{array}{cc|cc}-1&0&-2&-3\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).}
16734:
16158:
11981:
that uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as the
3078:{\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\1&-1&0&1\end{array}}\right).}
2883:
1086:
94:
49:
16819:
16749:
16713:
16584:
11931:
matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the
7638:{\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {x} _{0}&\mathbf {x} _{1}&\mathbf {x} _{2}\end{bmatrix}}}
233:
215:
160:
63:
4691:
1002:. (In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
17277:
12002:
10794:
4276:
3836:{\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {A} =\mathbf {I} .}
2725:
127:
17:
12983:) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.
11937:
11794:
11634:{\displaystyle \mathbf {W} _{3}=\mathbf {S} ^{-1}\mathbf {W} _{1}=\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},}
10475:
are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization
7304:
2327:
2269:
2084:
This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of
1015:. (Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
17930:
17239:
17175:
11982:
5252:
4178:
11535:{\displaystyle \mathbf {W} _{2}=\mathbf {W} _{1}\mathbf {C} ^{T}=\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T},}
3305:
17477:
12933:
11769:
together with some additions, subtractions, negations and transpositions of negligible complexity. Any matrix
5085:
2902:
is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an
2385:
98:
109:
16679:
16608:
3144:
413:
is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.
17694:
17331:
17017:
16889:
16669:
16124:
11978:
6410:
of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.
4169:
3516:
2850:{\displaystyle \mathbf {C} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\{\tfrac {2}{3}}&-1\end{pmatrix}}.}
1909:
17769:
17122:
16972:
16674:
16109:
13554:
4292:
13310:{\displaystyle \mathbf {e} _{i}=\mathbf {e} ^{i},\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}}
13010:
5622:
method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:
3848:
17925:
17447:
17027:
16921:
16703:
16231:
15971:
14643:
12205:. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a
3635:{\displaystyle \left({\begin{array}{cc|cc}1&0&2&3\\0&1&2&2\end{array}}\right).}
3467:
14058:
13670:
13641:
13199:
13163:
9267:
equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of
7706:
7677:
7648:
4121:{\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} ,}
3727:
1976:
18029:
17900:
17814:
17267:
16916:
15947:
13817:
13056:
10823:
6524:
1362:
2244:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\operatorname {adj} (\mathbf {A} ).}
18134:
18024:
17732:
17412:
17259:
17142:
16801:
16797:
16781:
16129:
16084:
10457:
10453:
8081:{\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}
6407:
5464:
2095:
14799:
14164:
11828:
11772:
8227:{\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}
2863:
2703:
733:
18283:
18169:
18098:
17980:
17840:
17437:
17324:
17305:
17234:
17012:
16882:
16843:
16119:
16027:
4652:
995:
928:
87:
14154:{\displaystyle (\mathbf {x} _{1}\wedge \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})=0}
13326:
11924:{\displaystyle \mathbf {M} ^{-1}=\left(\mathbf {M} ^{T}\mathbf {M} \right)^{-1}\mathbf {M} ^{T}}
10797:, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
18298:
18039:
17622:
17427:
17069:
17002:
16992:
16187:
16104:
11932:
5566:
5452:
4632:
389:
134:
55:
1060:
18293:
17985:
17722:
17572:
17567:
17402:
17377:
17372:
17084:
17079:
17074:
17007:
16952:
16359:
16202:
12729:
905:
661:
497:
373:
16869:
13611:
2985:{\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.}
2906:
is first created with the left side being the matrix to invert and the right side being the
2690:{\displaystyle \mathbf {B} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.}
2514:
18179:
17537:
17367:
17347:
17094:
17059:
17046:
16937:
16789:
16691:
16522:
16306:
16094:
9251:
6850:
5209:
4683:
4141:
3117:
3090:
2899:
1949:
673:
193:
479:
475:
8:
18200:
18174:
17752:
17557:
17547:
17272:
17152:
17127:
16977:
16162:
15943:
6132:
5607:
5583:
4267:
2438:
722:
669:
644:
624:
417:
16526:
16310:
16074:
to be invertible for the receiver to be able to figure out the transmitted information.
5171:{\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1},}
763:
is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a
18251:
18205:
18195:
18149:
18144:
18073:
18009:
17875:
17612:
17607:
17542:
17532:
17397:
16982:
16861:
16322:
16099:
16051:
15268:
13157:
12191:{\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }(\mathbf {I} -\mathbf {A} )^{n}.}
11375:{\displaystyle \mathbf {S} =\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T}}
4165:
2510:
1041:
636:
16384:
5546:{\displaystyle \mathbf {A} ^{-1}=\left(\mathbf {L} ^{*}\right)^{-1}\mathbf {L} ^{-1},}
18288:
18262:
18246:
18049:
18044:
18034:
18014:
17975:
17970:
17799:
17794:
17779:
17765:
17760:
17707:
17602:
17552:
17497:
17467:
17462:
17442:
17432:
17392:
17180:
17137:
16957:
16730:
16709:
16612:
16580:
16489:
16428:
16403:
16347:
16326:
16294:
16235:
16166:
16003:
13322:
13190:
5308:
3721:
3566:
This yields the identity matrix on the left side and the inverse matrix on the right:
2479:
2134:
1446:{\displaystyle (\mathbf {A} ^{\mathrm {T} })^{-1}=(\mathbf {A} ^{-1})^{\mathrm {T} }}
1132:
16577:
Matrix
Differential Calculus : with Applications in Statistics and Econometrics
18257:
18225:
18154:
18093:
18088:
18068:
18004:
17910:
17880:
17865:
17845:
17784:
17737:
17712:
17702:
17673:
17592:
17587:
17562:
17492:
17472:
17382:
17362:
17185:
17089:
16942:
16687:
16645:
16530:
16467:
16380:
16314:
16260:, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, Providence:
16089:
15982:
13318:
12992:
9230:
5601:
5563:
4767:
4150:, the right portion applied the same elementary row operation sequence will become
2903:
2427:
1037:
640:
17850:
14633:{\displaystyle \mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}}
17955:
17890:
17870:
17855:
17835:
17819:
17717:
17648:
17638:
17597:
17482:
17452:
17244:
17037:
16997:
16987:
16724:
16341:
13847:
13194:
12929:
9258:
5611:
5246:
4280:
3711:{\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}2&3\\2&2\end{pmatrix}}.}
2907:
2146:
868:
857:
835:
824:
369:
14048:{\displaystyle \mathbf {X} ^{-1}\mathbf {X} =\left=\left=\left=\mathbf {I} _{n}}
12201:
Truncating the sum results in an "approximate" inverse which may be useful as a
18215:
18159:
18139:
18124:
18083:
17960:
17920:
17885:
17809:
17748:
17727:
17668:
17658:
17643:
17577:
17522:
17512:
17507:
17417:
17249:
17170:
16905:
16699:
16650:
16633:
16039:
16014:, world-to-subspace-to-world object transformations, and physical simulations.
12202:
12107:
7997:
7561:
7295:
2446:
893:
632:
620:
433:
247:
16535:
16510:
16276:
Harvard
University Center for Research in Computing Technology Report TR-02-85
9273:
equals the nullity of the sub-block in the upper right of the inverse matrix.
4270:
of related matrices that behave enough like the sequence manufactured for the
18277:
18220:
18078:
18019:
17950:
17940:
17935:
17860:
17789:
17663:
17653:
17582:
17502:
17487:
17422:
17282:
17205:
17165:
17132:
17112:
16550:
16150:
16114:
12206:
11989:
shows that there exist matrix multiplication algorithms with a complexity of
8405:, the Cayley–Hamilton method leads to an expression that is still tractable:
7557:
4642:
2495:
2260:
453:
445:
258:
16371:
Tzon-Tzer, Lu; Sheng-Hua, Shiou (2002). "Inverses of 2 × 2 block matrices".
2766:
As an example of a non-invertible, or singular, matrix, consider the matrix
2592:{\displaystyle \mathbf {A} ={\begin{pmatrix}2&4\\2&4\end{pmatrix}}.}
18103:
18060:
17965:
17678:
17617:
17527:
17407:
17215:
17104:
17054:
16947:
16740:
16600:
11004:
If the given invertible matrix is a symmetric matrix with invertible block
9239:) is a small matrix, since they are the only matrices requiring inversion.
9182:
2430:
1886:(and vice versa interchanging rows for columns). To see this, suppose that
474:, do not have an inverse. However, in some cases such a matrix may have a
16471:
13807:{\displaystyle \mathbf {X} \mathbf {X} ^{-1}=\left=\left=\mathbf {I} _{n}}
17945:
17915:
17683:
17517:
17387:
17195:
17160:
17117:
16962:
16011:
16007:
11394:
and only 4 multiplications of half-sized matrices, if organized properly
10807:
6116:
5204:
4172:
may be convenient, if it is convenient to find a suitable starting seed:
2434:
2015:
1877:
1019:
616:
449:
441:
437:
16050:
receive antennas. The signal arriving at each receive antenna will be a
10800:
This formula simplifies significantly when the upper right block matrix
2994:
The first step to compute its inverse is to create the augmented matrix
1545:{\displaystyle (\mathbf {AB} )^{-1}=\mathbf {B} ^{-1}\mathbf {A} ^{-1}.}
17996:
17457:
17224:
16967:
16748:
Petersen, Kaare Brandt; Pedersen, Michael
Syskind (November 15, 2012).
16695:
16579:(Revised ed.). New York: John Wiley & Sons. pp. 151–152.
16318:
4256:
2509:
In practice however, one may encounter non-invertible matrices. And in
2491:
2450:
2442:
2085:
1048:
982:
16632:
Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan (2009).
11825:, which is exactly invertible (and positive definite), if and only if
6508:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det \mathbf {A} }}\left.}
5096:
can be eigendecomposed, and if none of its eigenvalues are zero, then
18230:
17804:
17022:
12090:{\displaystyle \lim _{n\to \infty }(\mathbf {I} -\mathbf {A} )^{n}=0}
6142:
5619:
4260:
2472:
1348:{\displaystyle (\mathbf {Ax} )^{+}=\mathbf {x} ^{+}\mathbf {A} ^{-1}}
978:
807:
14331:
To derive the above expression for the derivative of the inverse of
9210:
must be nonsingular.) This strategy is particularly advantageous if
2088:
vectors (but not necessarily orthonormal vectors) to the columns of
1144:
Furthermore, the following properties hold for an invertible matrix
76:
18164:
17190:
16800:
external links, and converting useful links where appropriate into
5249:
whose diagonal entries are the corresponding eigenvalues, that is,
4271:
2475:
2423:
628:
16511:"A p-adic algorithm for computing the inverse of integer matrices"
4838:{\displaystyle t_{l}=-(l-1)!\operatorname {tr} \left(A^{l}\right)}
623:
numbers, all these definitions can be given for matrices over any
17316:
16874:
16847:
16463:
16017:
12925:
9247:
4263:
have done work that includes ways of generating a starting seed.
14672:
gives the correct expression for the derivative of the inverse:
4036:{\displaystyle \mathbf {I} \mathbf {A} ^{-1}=\mathbf {A} ^{-1},}
1854:{\displaystyle \det \mathbf {A} ^{-1}=(\det \mathbf {A} )^{-1}.}
17200:
16010:
rendering and 3D simulations. Examples include screen-to-world
12937:
11968:
satisfies the invertibility condition for its left upper block
11442:{\displaystyle \mathbf {W} _{1}=\mathbf {C} \mathbf {A} ^{-1},}
2377:
6161:
matrices. Inversion of these matrices can be done as follows:
5356:{\displaystyle \mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }.}
2433:. This is true because singular matrices are the roots of the
16708:(2nd ed.). MIT Press and McGraw-Hill. pp. 755–760.
14640:
from both sides of the above and multiplying on the right by
14335:, one can differentiate the definition of the matrix inverse
343:{\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},}
12963:
matrix multiplication is used. The method relies on solving
1275:{\displaystyle (k\mathbf {A} )^{-1}=k^{-1}\mathbf {A} ^{-1}}
16023:
15997:
5614:, can also be an efficient way to calculate the inverse of
16686:
14373:{\displaystyle \mathbf {A} ^{-1}\mathbf {A} =\mathbf {I} }
11791:
has an associated positive semidefinite, symmetric matrix
9242:
This technique was reinvented several times and is due to
8392:
With increasing dimension, expressions for the inverse of
8142:{\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} }
15942:
Some of the properties of inverse matrices are shared by
13667:" is removed from that place in the above expression for
9191:
must be square, so that it can be inverted. Furthermore,
4266:
Newton's method is particularly useful when dealing with
2602:
We can see the rank of this 2-by-2 matrix is 1, which is
2490:
matrices. Equivalently, the set of singular matrices is
2098:
to this initial set to determine the rows of the inverse
11382:. This requires 2 inversions of the half-sized matrices
5369:
is a diagonal matrix, its inverse is easy to calculate:
2077:{\displaystyle v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.}
1595:{\displaystyle \mathbf {A} _{1},\dots ,\mathbf {A} _{k}}
1044:(i.e. multiplicatively invertible element) of that ring.
977:. (Here, "bijective" can equivalently be replaced with "
615:
While the most common case is that of matrices over the
16450:
T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein,
16292:
13149:{\displaystyle \mathbf {x} _{i}=x^{ij}\mathbf {e} _{j}}
12943:
approximation method converges to an exact solution in
9254:(1937), who generalized it and proved its correctness.
16022:
Matrix inversion also plays a significant role in the
11091:
11028:
10898:
10842:
10689:
10549:
10493:
9399:
9343:
8805:
8749:
8098:
is orthogonal to the non-corresponding two columns of
7788:
7584:
6769:
6663:
6564:
6335:
6260:
6195:
6051:
5988:
5728:
5705:
5660:
3674:
3420:
3258:
3227:
3018:
2945:
2933:
2817:
2801:
2789:
2728:
2650:
2638:
2555:
2445:
in the entries of the matrix. Thus in the language of
2094:
are known. In which case, one can apply the iterative
432:. A square matrix with entries in a field is singular
16038:
receive antennas. Unique signals, occupying the same
15599:
15294:
15271:
15115:
14973:
14825:
14802:
14681:
14646:
14588:
14393:
14341:
14216:
14167:
14090:
14061:
13856:
13820:
13702:
13673:
13644:
13614:
13557:
13338:
13231:
13202:
13166:
13105:
13059:
13013:
12741:
12593:
12410:
12218:
12119:
12106:
is nonsingular and its inverse may be expressed by a
12038:
11940:
11853:
11831:
11797:
11775:
11646:
11547:
11454:
11400:
11323:
11021:
10835:
10486:
9775:
9336:
9282:) performed matrix block operations that operated on
8742:
8414:
8259:
8158:
8110:
8009:
7741:
7709:
7680:
7651:
7570:
7374:
7307:
6874:
6539:
6422:
6170:
5941:
5631:
5476:
5436:{\displaystyle \left_{ii}={\frac {1}{\lambda _{i}}}.}
5378:
5317:
5255:
5212:
5111:
4854:
4776:
4694:
4655:
4324:
4181:
4052:
3991:
3886:
3851:
3759:
3730:
3650:
3572:
3519:
3470:
3374:
3308:
3209:
3147:
3120:
3093:
3000:
2919:
2866:
2775:
2706:
2624:
2541:
2388:
2330:
2272:
2254:
2179:
2024:
1979:
1952:
1912:
1802:
1616:
1558:
1481:
1379:
1293:
1216:
1202:{\displaystyle (\mathbf {A} ^{-1})^{-1}=\mathbf {A} }
1159:
1089:
1063:
736:
302:
2368:
Over the field of real numbers, the set of singular
16295:"Superconducting quark matter in SU(2) color group"
14186:
8731:by using the following analytic inversion formula:
7556:inverse can be expressed concisely in terms of the
2722:is invertible. To check this, one can compute that
660:invertible matrices together with the operation of
452:that the matrix is singular is 0, that is, it will
188:
may be too technical for most readers to understand
101:. Unsourced material may be challenged and removed.
16370:
15978:necessary that the matrix involved be invertible.
15926:
15579:
15277:
15254:
15095:
14953:
14808:
14785:
14664:
14632:
14571:
14372:
14320:
14175:
14153:
14076:
14047:
13838:
13806:
13688:
13659:
13630:
13600:
13540:
13309:
13217:
13181:
13148:
13083:
13045:
12986:
12899:
12707:
12567:
12362:
12190:
12089:
11960:
11923:
11839:
11817:
11783:
11761:
11633:
11534:
11441:
11374:
11293:
10993:
10769:
10422:
9728:
9133:
8711:
8376:
8226:
8141:
8080:
7944:
7724:
7695:
7666:
7637:
7541:
7354:
7277:
6838:
6507:
6383:
6094:
5921:
5545:
5435:
5355:
5287:
5225:
5170:
5061:
4837:
4766:The formula can be rewritten in terms of complete
4755:
4674:
4602:
4245:
4120:
4035:
3977:
3872:
3835:
3745:
3710:
3634:
3558:
3505:
3454:
3360:
3302:Next, subtract row 2, multiplied by 3, from row 1
3292:
3195:
3133:
3106:
3077:
2984:
2874:
2849:
2755:
2714:
2689:
2591:
2412:
2352:
2294:
2243:
2076:
2006:
1965:
1932:
1853:
1787:
1594:
1544:
1445:
1347:
1274:
1201:
1108:
1069:
1040:is invertible if and only if its determinant is a
744:
342:
16784:may not follow Knowledge's policies or guidelines
16508:
16454:, 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2.
12000:operations, while the best proven lower bound is
8250:to be unity. For example, the first diagonal is:
6347:
6346:
6272:
6271:
2107:A matrix that is its own inverse (i.e., a matrix
1109:{\displaystyle \mathbf {A} -\lambda \mathbf {I} }
18275:
16747:
16515:Journal of Computational and Applied Mathematics
16293:Kondratyuk, L. A.; Krivoruchenko, M. I. (1992).
16185:
16026:(Multiple-Input, Multiple-Output) technology in
12499:
12412:
12040:
8439:
8159:
8010:
7766:
7399:
7308:
6747:
6640:
6447:
6244:
4879:
4349:
2729:
2204:
1827:
1803:
702:
16836:"Inverse Matrices, Column Space and Null Space"
16726:Matrix Mathematics: Theory, Facts, and Formulas
16574:
10810:. This formulation is useful when the matrices
4277:matrix square roots by Denman–Beavers iteration
16018:Matrix inverses in MIMO wireless communication
13325:) we compute the reciprocal (sometimes called
11987:Research into matrix multiplication complexity
9276:The inversion procedure that led to Equation (
8387:
6518:
6148:
4756:{\displaystyle s+\sum _{l=1}^{n-1}lk_{l}=n-1.}
2889:
2756:{\textstyle \det \mathbf {B} =-{\frac {1}{2}}}
2140:
17332:
16890:
16373:Computers & Mathematics with Applications
16258:Efficient Parallel Solution of Linear Systems
16002:Matrix inversion plays a significant role in
12382:matrix multiplications are needed to compute
11977:These formulas together allow to construct a
6157:listed above yields the following result for
4128:we create the augumented matrix by combining
16857:"Linear Algebra Lecture on Inverse Matrices"
16846:from the original on 2021-11-03 – via
16729:(2nd ed.). Princeton University Press.
16631:
16226:Horn, Roger A.; Johnson, Charles R. (1985).
11961:{\displaystyle \mathbf {M} ^{T}\mathbf {M} }
11818:{\displaystyle \mathbf {M} ^{T}\mathbf {M} }
10822:have relatively simple inverse formulas (or
7355:{\displaystyle \det(\mathbf {A} )=aA+bB+cC.}
2353:{\displaystyle \mathbf {BA} =\mathbf {I} \ }
2295:{\displaystyle \mathbf {AB} =\mathbf {I} \ }
1361:has orthonormal columns, where denotes the
16225:
15988:
5288:{\displaystyle \Lambda _{ii}=\lambda _{i}.}
4246:{\displaystyle X_{k+1}=2X_{k}-X_{k}AX_{k}.}
64:Learn how and when to remove these messages
17906:Fundamental (linear differential equation)
17339:
17325:
16897:
16883:
16427:. Princeton University Press. p. 45.
16402:. Princeton University Press. p. 44.
16291:A proof can be found in the Appendix B of
15994:a matrix inverse are known in many cases.
11010:the following block inverse formula holds
7732:) is invertible, its inverse is given by
5102:is invertible and its inverse is given by
3361:{\displaystyle (R_{1}-3\,R_{2}\to R_{1}),}
16833:
16820:Learn how and when to remove this message
16722:
16649:
16611:(Third ed.), Springer, p. 446,
16575:Magnus, Jan R.; Neudecker, Heinz (1999).
16534:
16446:
16444:
16422:
16397:
14947:
13205:
9246:(1923), who used it for the inversion of
6824:
6818:
6812:
6804:
6798:
6792:
6784:
6778:
6772:
6718:
6712:
6706:
6698:
6692:
6686:
6678:
6672:
6666:
6366:
6340:
6339:
6338:
6291:
6265:
6264:
6263:
5446:
4286:
3526:
3328:
2413:{\displaystyle \mathbb {R} ^{n\times n},}
2391:
738:
234:Learn how and when to remove this message
216:Learn how and when to remove this message
200:, without removing the technical details.
161:Learn how and when to remove this message
27:Matrix which has a multiplicative inverse
15998:Matrix inverses in real-time simulations
15970:necessary to invert a matrix to solve a
14199:. Then the derivative of the inverse of
7365:The Cayley–Hamilton decomposition gives
2913:For example, take the following matrix:
1131:can be expressed as a finite product of
639:). However, in the case of a ring being
372:and the multiplication used is ordinary
18211:Matrix representation of conic sections
16638:Communications in Mathematical Sciences
16486:Matrix Algorithms: Basic decompositions
16483:
16273:
16255:
16046:transmit antennas and are received via
15966:For most practical applications, it is
12910:
5072:This is described in more detail under
3196:{\displaystyle (R_{1}+R_{2}\to R_{2}).}
2894:
456:be singular. Non-square matrices, i.e.
376:. If this is the case, then the matrix
14:
18276:
17288:Comparison of linear algebra libraries
16441:
16339:
15937:
8722:
6856:is not to be confused with the matrix
5467:, then its inverse can be obtained as
2615:Consider the following 2-by-2 matrix:
1036:. (In general, a square matrix over a
17320:
16878:
16702:(2001) . "28.4: Inverting matrices".
16599:
16200:
16149:
15974:; however, for a unique solution, it
13551:as the columns of the inverse matrix
12967:linear systems via Dixon's method of
7645:(consisting of three column vectors,
5079:
198:make it understandable to non-experts
16834:Sanderson, Grant (August 15, 2016).
16764:
16509:Haramoto, H.; Matsumoto, M. (2009).
14183:is not invertible (has no inverse).
12018:
11012:
10477:
9766:
9327:
9306:are operated on first, and provided
8733:
7969:, is equal to the triple product of
5595:
3845:Applying right-multiplication using
3559:{\displaystyle (2\,R_{2}\to R_{2}).}
1933:{\displaystyle v_{i}^{\mathrm {T} }}
957:is bijective; that is, the equation
172:
99:adding citations to reliable sources
70:
29:
14191:Suppose that the invertible matrix
14084:are not linearly independent, then
13601:{\displaystyle \mathbf {X} ^{-1}=.}
8104:(causing the off-diagonal terms of
2886:for a matrix to be non-invertible.
2155:can be used to find the inverse of
1139:
583:), then it has a right inverse, an
24:
17346:
16904:
16854:
16662:
16346:(3rd ed.). SIAM. p. 71.
16159:Undergraduate Texts in Mathematics
15894:
15725:
15527:
15515:
15455:
15433:
15394:
15382:
15322:
15303:
15226:
15059:
15047:
14999:
14978:
14926:
14755:
14743:
14708:
14686:
14620:
14608:
14548:
14536:
14519:
14507:
14470:
14448:
14431:
14398:
14380:and then solve for the inverse of
14290:
14278:
14243:
14221:
13046:{\displaystyle \mathbf {X} =\left}
12932:entries and we seek a solution in
12628:
12584:is nonsingular and its inverse is
12509:
12488:
12485:
12422:
12154:
12050:
7925:
7877:
7829:
6732:
6019:
5686:
5385:
5344:
5257:
4159:
3873:{\displaystyle \mathbf {A} ^{-1},}
3087:Call the first row of this matrix
2884:necessary and sufficient condition
2255:In relation to the identity matrix
2036:
1924:
1437:
1391:
967:has exactly one solution for each
945:The linear transformation mapping
791:The linear transformation mapping
25:
18310:
16760:
14665:{\displaystyle \mathbf {A} ^{-1}}
6413:The Cayley–Hamilton method gives
3506:{\displaystyle (-R_{1}\to R_{1})}
2263:of matrix multiplication that if
45:This article has multiple issues.
18245:
17301:
17300:
17278:Basic Linear Algebra Subprograms
17036:
16769:
16274:Pan, Victor; Reif, John (1985),
16256:Pan, Victor; Reif, John (1985),
15950:), which can be defined for any
15858:
15852:
15838:
15796:
15770:
15759:
15704:
15698:
15681:
15642:
15619:
15608:
15539:
15520:
15498:
15439:
15406:
15387:
15362:
15309:
15214:
15196:
15188:
15151:
15134:
15123:
15083:
15052:
15036:
14989:
14908:
14902:
14888:
14867:
14844:
14833:
14767:
14748:
14726:
14692:
14649:
14613:
14591:
14562:
14541:
14512:
14490:
14481:
14454:
14421:
14407:
14366:
14358:
14344:
14302:
14283:
14261:
14227:
14187:Derivative of the matrix inverse
14169:
14132:
14111:
14096:
14077:{\displaystyle \mathbf {x} _{i}}
14064:
14035:
13989:
13974:
13944:
13929:
13907:
13892:
13873:
13859:
13794:
13748:
13733:
13710:
13704:
13689:{\displaystyle \mathbf {x} ^{i}}
13676:
13660:{\displaystyle \mathbf {x} _{i}}
13647:
13560:
13515:
13494:
13476:
13455:
13412:
13369:
13341:
13279:
13264:
13249:
13234:
13218:{\displaystyle \mathbb {R} ^{n}}
13182:{\displaystyle \mathbf {e} _{j}}
13169:
13136:
13108:
13015:
12876:
12868:
12851:
12817:
12808:
12800:
12783:
12762:
12744:
12686:
12666:
12658:
12641:
12596:
12535:
12529:
12521:
12457:
12443:
12434:
12395:is "near" the invertible matrix
12334:
12326:
12315:
12265:
12257:
12171:
12163:
12122:
12067:
12059:
11954:
11943:
11911:
11891:
11880:
11856:
11833:
11811:
11800:
11777:
11743:
11737:
11723:
11711:
11696:
11681:
11664:
11649:
11615:
11609:
11595:
11580:
11565:
11550:
11519:
11504:
11498:
11484:
11472:
11457:
11423:
11417:
11403:
11362:
11347:
11341:
11333:
11325:
11267:
11250:
11244:
11230:
11208:
11196:
11181:
11161:
11155:
11141:
11129:
11114:
11096:
11062:
11055:
11040:
11032:
10967:
10950:
10947:
10932:
10919:
10903:
10869:
10862:
10853:
10846:
10752:
10736:
10730:
10709:
10703:
10693:
10658:
10644:
10638:
10630:
10617:
10608:
10587:
10573:
10567:
10559:
10520:
10513:
10504:
10497:
10398:
10384:
10381:
10372:
10345:
10342:
10322:
10308:
10305:
10296:
10285:
10271:
10253:
10234:
10231:
10211:
10197:
10194:
10185:
10153:
10139:
10136:
10127:
10116:
10102:
10078:
10064:
10061:
10052:
10041:
10027:
10005:
10002:
9982:
9968:
9965:
9956:
9932:
9929:
9909:
9895:
9892:
9883:
9872:
9858:
9840:
9813:
9799:
9796:
9787:
9702:
9699:
9679:
9665:
9662:
9653:
9642:
9628:
9610:
9587:
9573:
9570:
9561:
9550:
9536:
9514:
9511:
9491:
9477:
9474:
9465:
9435:
9421:
9418:
9409:
9370:
9363:
9354:
9347:
9102:
9088:
9085:
9076:
9054:
9051:
9031:
9017:
9014:
9005:
8973:
8959:
8956:
8947:
8936:
8922:
8902:
8899:
8879:
8865:
8862:
8853:
8842:
8828:
8810:
8776:
8769:
8760:
8753:
8691:
8682:
8665:
8642:
8614:
8595:
8577:
8558:
8528:
8513:
8486:
8446:
8417:
8358:
8343:
8329:
8325:
8307:
8292:
8278:
8274:
8211:
8196:
8178:
8166:
8135:
8121:
8112:
8062:
8047:
8029:
8017:
8000:formed by the rows or columns:
7909:
7894:
7861:
7846:
7813:
7798:
7773:
7744:
7725:{\displaystyle \mathbf {x} _{2}}
7712:
7696:{\displaystyle \mathbf {x} _{1}}
7683:
7667:{\displaystyle \mathbf {x} _{0}}
7654:
7617:
7603:
7589:
7572:
7521:
7512:
7501:
7493:
7474:
7446:
7406:
7377:
7315:
7294:can be computed by applying the
6754:
6647:
6542:
6493:
6485:
6475:
6451:
6425:
6248:
6173:
6075:
6055:
6013:
5992:
5949:
5898:
5876:
5859:
5818:
5799:
5785:
5766:
5747:
5733:
5709:
5680:
5664:
5634:
5527:
5502:
5479:
5338:
5320:
5152:
5137:
5131:
5114:
4919:
4886:
4857:
4570:
4395:
4356:
4327:
4170:multiplicative inverse algorithm
4111:
4100:
4088:
4067:
4055:
4043:which is the inverse we want.
4017:
3999:
3993:
3959:
3953:
3945:
3934:
3922:
3901:
3889:
3854:
3826:
3818:
3807:
3795:
3774:
3762:
3746:{\displaystyle \mathbf {E} _{n}}
3733:
3653:
2921:
2868:
2777:
2733:
2708:
2626:
2543:
2343:
2335:
2332:
2285:
2277:
2274:
2231:
2211:
2182:
2007:{\displaystyle 1\leq i,j\leq n.}
1831:
1808:
1764:
1744:
1715:
1695:
1667:
1649:
1634:
1622:
1582:
1561:
1526:
1511:
1489:
1486:
1418:
1385:
1332:
1320:
1301:
1298:
1259:
1224:
1195:
1165:
1102:
1091:
327:
318:
315:
307:
304:
177:
75:
34:
18113:Used in science and engineering
17176:Seven-dimensional cross product
16625:
16593:
16568:
16543:
16502:
16477:
16457:
16416:
16391:
16186:J.-S. Roger Jang (March 2001).
16165:(published 2015). p. 296.
16070:. It is crucial for the matrix
16058:transmitted signals forming an
15961:
13839:{\displaystyle \delta _{i}^{j}}
13084:{\displaystyle 1\leq i,j\leq n}
12987:Reciprocal basis vectors method
11983:matrix multiplication algorithm
9607:
9325:are nonsingular, the result is
8237:causes the diagonal entries of
7230:
7184:
7096:
7053:
6965:
6919:
1864:The rows of the inverse matrix
86:needs additional citations for
53:or discuss these issues on the
17356:Explicitly constrained entries
16551:"IML - Integer Matrix Library"
16364:
16343:Introduction to linear algebra
16333:
16285:
16267:
16249:
16219:
16194:
16188:"Matrix Inverse in Block Form"
16179:
16143:
16030:. The MIMO system consists of
15981:Decomposition techniques like
15884:
15866:
15775:
15755:
15624:
15604:
15565:
15547:
15218:
15210:
15192:
15184:
15155:
15147:
15138:
15119:
15087:
15079:
15040:
15032:
14993:
14985:
14944:
14931:
14425:
14402:
14142:
14091:
14055:, as required. If the vectors
13619:
13615:
13592:
13576:
13526:
13471:
13465:
13435:
13431:
13407:
13392:
13382:
12880:
12864:
12812:
12796:
12670:
12654:
12506:
12419:
12339:
12322:
12270:
12253:
12176:
12159:
12072:
12055:
12047:
8652:
8637:
8619:
8604:
8568:
8553:
8538:
8523:
8491:
8476:
8450:
8442:
8368:
8338:
8317:
8287:
8221:
8191:
8170:
8162:
8072:
8042:
8021:
8013:
7919:
7889:
7871:
7841:
7823:
7793:
7777:
7769:
7484:
7469:
7451:
7436:
7410:
7402:
7319:
7311:
7265:
7247:
7222:
7204:
7176:
7158:
7134:
7116:
7088:
7070:
7045:
7027:
7000:
6982:
6957:
6939:
6911:
6893:
6758:
6750:
6651:
6643:
5086:Eigendecomposition of a matrix
5053:
5002:
4977:
4965:
4948:
4938:
4890:
4882:
4805:
4793:
4500:
4490:
4360:
4352:
3550:
3537:
3520:
3500:
3487:
3471:
3464:Finally, multiply row 1 by −1
3352:
3339:
3309:
3187:
3174:
3148:
2235:
2227:
2215:
2207:
2170:is an invertible matrix, then
1836:
1824:
1678:
1617:
1494:
1482:
1432:
1413:
1398:
1380:
1306:
1294:
1229:
1217:
1179:
1160:
13:
1:
18130:Fundamental (computer vision)
16723:Bernstein, Dennis S. (2009).
16609:Graduate Texts in Mathematics
16385:10.1016/S0898-1221(01)00278-4
16136:
12971:-adic approximation (each in
10456:, which is equivalent to the
6530:matrix inversion is given by
5606:Writing the transpose of the
4281:imperfect computer arithmetic
703:The invertible matrix theorem
697:
17018:Eigenvalues and eigenvectors
16870:Moore-Penrose Inverse Matrix
16280:Aiken Computation Laboratory
16125:Singular value decomposition
14809:{\displaystyle \varepsilon }
14176:{\displaystyle \mathbf {X} }
12718:If it is also the case that
11979:divide and conquer algorithm
11840:{\displaystyle \mathbf {M} }
11784:{\displaystyle \mathbf {M} }
10795:Weinstein–Aronszajn identity
4301:to be expressed in terms of
2875:{\displaystyle \mathbf {C} }
2715:{\displaystyle \mathbf {B} }
1057:. (More generally, a number
745:{\displaystyle \mathbb {R} }
7:
17896:Duplication and elimination
17695:eigenvalues or eigenvectors
16675:Encyclopedia of Mathematics
16203:"Invertible Matrix Theorem"
16110:Partial inverse of a matrix
16077:
11307:
10783:
10448:
10436:
9760:
9754:
9742:
9278:
9147:
8388:Inversion of 4 × 4 matrices
6519:Inversion of 3 × 3 matrices
6149:Inversion of 2 × 2 matrices
4675:{\displaystyle k_{l}\geq 0}
3141:. Then, add row 1 to row 2
2890:Methods of matrix inversion
2612:, so it is non-invertible.
2532:is a non-invertible matrix
2520:
2141:In relation to its adjugate
10:
18315:
17829:With specific applications
17458:Discrete Fourier Transform
16705:Introduction to Algorithms
16651:10.4310/CMS.2009.v7.n3.a12
16452:Introduction to Algorithms
16423:Bernstein, Dennis (2005).
16398:Bernstein, Dennis (2005).
16232:Cambridge University Press
15972:system of linear equations
12990:
12732:1 then this simplifies to
11847:is invertible. By writing
5599:
5450:
5083:
2471:invertible matrices are a
2376:matrices, considered as a
2363:
420:, a square matrix that is
382:is uniquely determined by
18239:
18188:
18120:Cabibbo–Kobayashi–Maskawa
18112:
18058:
17994:
17828:
17747:Satisfying conditions on
17746:
17692:
17631:
17355:
17296:
17258:
17214:
17151:
17103:
17045:
17034:
16930:
16912:
16840:Essence of Linear Algebra
16536:10.1016/j.cam.2008.07.044
16484:Stewart, Gilbert (1998).
16155:Linear Algebra Done Right
15265:Given a positive integer
12209:. As such, it satisfies
11985:that is used internally.
9261:says that the nullity of
6525:computationally efficient
6394:This is possible because
2460:matrices are invertible.
16340:Strang, Gilbert (2003).
16299:Zeitschrift für Physik A
16130:Woodbury matrix identity
16085:Binomial inverse theorem
15989:Regression/least squares
11933:positive definite matrix
10458:binomial inverse theorem
10454:Woodbury matrix identity
4645:. The sum is taken over
4641:given by the sum of the
2525:An example with rank of
1122:is the identity matrix.)
1070:{\displaystyle \lambda }
815:is an invertible matrix.
17478:Generalized permutation
16670:"Inversion of a matrix"
16605:Advanced Linear Algebra
16120:Rybicki Press algorithm
16028:wireless communications
14816:is a small number then
14195:depends on a parameter
8149:be zero). Dividing by
5307:is guaranteed to be an
4309:, traces and powers of
4293:Cayley–Hamilton theorem
2016:Euclidean inner product
1047:The number 0 is not an
528:has a left inverse, an
18252:Mathematics portal
17003:Row and column vectors
16105:Minor (linear algebra)
15928:
15835:
15678:
15581:
15495:
15359:
15279:
15256:
15097:
14955:
14810:
14787:
14666:
14634:
14573:
14374:
14322:
14177:
14155:
14078:
14049:
13840:
13808:
13690:
13661:
13632:
13631:{\displaystyle ()_{i}}
13608:Note that, the place "
13602:
13542:
13311:
13219:
13183:
13150:
13085:
13047:
12901:
12709:
12632:
12569:
12364:
12308:
12252:
12192:
12158:
12091:
12029:has the property that
11962:
11925:
11841:
11819:
11785:
11763:
11635:
11536:
11443:
11376:
11295:
10995:
10771:
10424:
9730:
9135:
8713:
8378:
8228:
8143:
8082:
7946:
7726:
7697:
7668:
7639:
7543:
7356:
7279:
6840:
6509:
6385:
6141:represents the matrix
6096:
5923:
5567:Cholesky decomposition
5547:
5453:Cholesky decomposition
5447:Cholesky decomposition
5437:
5357:
5289:
5227:
5172:
5074:Cayley–Hamilton method
5063:
4916:
4839:
4757:
4727:
4682:satisfying the linear
4676:
4604:
4486:
4392:
4295:allows the inverse of
4287:Cayley–Hamilton method
4247:
4122:
4037:
3979:
3874:
3837:
3747:
3712:
3636:
3560:
3507:
3456:
3362:
3294:
3197:
3135:
3108:
3079:
2986:
2876:
2851:
2763:, which is non-zero.
2757:
2716:
2691:
2593:
2511:numerical calculations
2414:
2354:
2296:
2245:
2078:
2008:
1967:
1934:
1855:
1789:
1596:
1546:
1447:
1349:
1276:
1203:
1110:
1071:
746:
664:and entries from ring
344:
17008:Row and column spaces
16953:Scalar multiplication
16750:"The Matrix Cookbook"
16692:Leiserson, Charles E.
16472:10.1145/509907.509932
16207:mathworld.wolfram.com
15948:Moore–Penrose inverse
15929:
15815:
15658:
15582:
15475:
15339:
15280:
15257:
15098:
14956:
14811:
14788:
14667:
14635:
14574:
14375:
14323:
14178:
14156:
14079:
14050:
13841:
13809:
13691:
13662:
13633:
13603:
13543:
13312:
13220:
13184:
13151:
13086:
13048:
12902:
12710:
12612:
12570:
12365:
12282:
12219:
12193:
12138:
12092:
11963:
11926:
11842:
11820:
11786:
11764:
11636:
11537:
11444:
11377:
11296:
10996:
10772:
10425:
9731:
9136:
8727:Matrices can also be
8714:
8398:get complicated. For
8379:
8229:
8144:
8083:
7947:
7727:
7698:
7669:
7640:
7544:
7357:
7280:
6841:
6510:
6386:
6097:
5924:
5548:
5438:
5363:Furthermore, because
5358:
5290:
5228:
5226:{\displaystyle q_{i}}
5173:
5064:
4896:
4840:
4758:
4701:
4677:
4605:
4460:
4366:
4248:
4123:
4038:
3980:
3875:
3838:
3748:
3713:
3637:
3561:
3508:
3457:
3363:
3295:
3198:
3136:
3134:{\displaystyle R_{2}}
3109:
3107:{\displaystyle R_{1}}
3080:
2987:
2877:
2852:
2758:
2717:
2692:
2594:
2415:
2355:
2297:
2246:
2079:
2009:
1968:
1966:{\displaystyle u_{j}}
1935:
1856:
1790:
1597:
1547:
1448:
1363:Moore–Penrose inverse
1350:
1277:
1204:
1111:
1072:
747:
662:matrix multiplication
424:invertible is called
374:matrix multiplication
345:
282:) if there exists an
17143:Gram–Schmidt process
17095:Gaussian elimination
16790:improve this article
16488:. SIAM. p. 55.
16153:(18 December 2014).
16095:Matrix decomposition
16066:transmission matrix
15944:generalized inverses
15597:
15292:
15269:
15113:
14971:
14823:
14800:
14679:
14644:
14586:
14391:
14339:
14214:
14165:
14088:
14059:
13854:
13818:
13700:
13671:
13642:
13612:
13555:
13336:
13229:
13200:
13164:
13103:
13095:rows interpreted as
13057:
13011:
12955:, assuming standard
12739:
12591:
12408:
12216:
12117:
12036:
11938:
11851:
11829:
11795:
11773:
11644:
11545:
11452:
11398:
11321:
11019:
10833:
10484:
9773:
9752:Equating Equations (
9334:
9252:Tadeusz Banachiewicz
9185:of arbitrary size. (
8740:
8412:
8257:
8156:
8108:
8007:
7739:
7707:
7678:
7649:
7568:
7372:
7305:
6872:
6537:
6420:
6168:
5939:
5629:
5474:
5376:
5315:
5253:
5210:
5109:
4852:
4774:
4692:
4684:Diophantine equation
4653:
4649:and the sets of all
4322:
4179:
4164:A generalization of
4142:Gaussian elimination
4050:
3989:
3884:
3849:
3757:
3728:
3648:
3570:
3517:
3468:
3372:
3306:
3207:
3145:
3118:
3091:
2998:
2917:
2900:Gaussian elimination
2895:Gaussian elimination
2864:
2773:
2726:
2704:
2622:
2539:
2437:function. This is a
2386:
2328:
2270:
2259:It follows from the
2177:
2096:Gram–Schmidt process
2022:
1977:
1950:
1910:
1800:
1614:
1556:
1479:
1377:
1291:
1214:
1157:
1087:
1077:is an eigenvalue of
1061:
734:
674:general linear group
388:, and is called the
300:
95:improve this article
18201:Linear independence
17448:Diagonally dominant
17273:Numerical stability
17153:Multilinear algebra
17128:Inner product space
16978:Linear independence
16802:footnote references
16527:2009JCoAM.225..320H
16311:1992ZPhyA.344...99K
16201:Weisstein, Eric W.
16163:Springer Publishing
15938:Generalized inverse
14964:More generally, if
14025:
13835:
13784:
13306:
13160:assumed) where the
12934:arbitrary-precision
12914:-adic approximation
12389:More generally, if
11678:
9294:first. Instead, if
8723:Blockwise inversion
7996:—the volume of the
7955:The determinant of
7288:The determinant of
6133:matrix of cofactors
5618:matrices, but this
5608:matrix of cofactors
5584:conjugate transpose
3985:And the right side
3722:elementary matrices
3114:and the second row
2860:The determinant of
2439:continuous function
2123:, and consequently
2041:
1940:and the columns of
1929:
1781:
1761:
1738:
1712:
1552:More generally, if
1455:For any invertible
1282:for nonzero scalar
1133:elementary matrices
1116:is singular, where
645:noncommutative ring
625:algebraic structure
464:matrices for which
110:"Invertible matrix"
18206:Matrix exponential
18196:Jordan normal form
18030:Fisher information
17901:Euclidean distance
17815:Totally unimodular
16983:Linear combination
16862:MIT OpenCourseWare
16425:Matrix Mathematics
16400:Matrix Mathematics
16360:Chapter 2, page 71
16319:10.1007/BF01291027
16100:Matrix square root
16052:linear combination
16006:, particularly in
15946:(for example, the
15924:
15922:
15577:
15575:
15275:
15252:
15173:
15093:
15021:
14951:
14806:
14783:
14662:
14630:
14569:
14370:
14318:
14173:
14151:
14074:
14045:
14011:
13836:
13821:
13804:
13770:
13686:
13657:
13638:" indicates that "
13628:
13598:
13538:
13329:) column vectors:
13307:
13292:
13215:
13179:
13158:Einstein summation
13146:
13081:
13043:
12936:rationals, then a
12897:
12705:
12565:
12513:
12426:
12401:in the sense that
12386:terms of the sum.
12360:
12188:
12087:
12054:
11958:
11921:
11837:
11815:
11781:
11759:
11662:
11631:
11532:
11439:
11372:
11291:
11282:
11068:
10991:
10982:
10875:
10767:
10758:
10678:
10526:
10420:
10418:
9726:
9717:
9376:
9131:
9122:
8782:
8729:inverted blockwise
8709:
8374:
8224:
8139:
8078:
7942:
7933:
7722:
7693:
7664:
7635:
7629:
7539:
7352:
7275:
7273:
6836:
6830:
6724:
6616:
6505:
6381:
6372:
6297:
6220:
6092:
6061:
5998:
5919:
5913:
5715:
5670:
5543:
5433:
5353:
5285:
5223:
5168:
5080:Eigendecomposition
5059:
4835:
4753:
4672:
4600:
4459:
4243:
4118:
4033:
3975:
3870:
3833:
3743:
3708:
3699:
3632:
3623:
3556:
3503:
3452:
3443:
3429:
3358:
3290:
3281:
3267:
3236:
3193:
3131:
3104:
3075:
3066:
3027:
2982:
2973:
2954:
2872:
2847:
2838:
2826:
2810:
2753:
2712:
2687:
2678:
2659:
2589:
2580:
2410:
2350:
2292:
2241:
2074:
2025:
2014:Then clearly, the
2004:
1963:
1930:
1913:
1900:where the rows of
1880:to the columns of
1851:
1785:
1762:
1742:
1713:
1693:
1592:
1542:
1443:
1345:
1272:
1199:
1106:
1067:
742:
340:
18271:
18270:
18263:Category:Matrices
18135:Fuzzy associative
18025:Doubly stochastic
17733:Positive-definite
17413:Block tridiagonal
17314:
17313:
17181:Geometric algebra
17138:Kronecker product
16973:Linear projection
16958:Vector projection
16855:Strang, Gilbert.
16830:
16829:
16822:
16755:. pp. 17–23.
16696:Rivest, Ronald L.
16688:Cormen, Thomas H.
16618:978-0-387-72828-5
16495:978-0-89871-414-2
16434:978-0-691-11802-4
16409:978-0-691-11802-4
16353:978-0-9614088-9-3
16278:, Cambridge, MA:
16241:978-0-521-38632-6
16172:978-3-319-11079-0
16004:computer graphics
15535:
15463:
15402:
15330:
15278:{\displaystyle n}
15164:
15067:
15012:
15007:
14763:
14716:
14628:
14556:
14527:
14478:
14439:
14298:
14251:
13491:
13323:geometric algebra
13191:orthonormal basis
12924:is a matrix with
12893:
12889:
12701:
12498:
12496:
12493:
12484:
12481:
12411:
12039:
12019:By Neumann series
11315:
11314:
10791:
10790:
10444:
10443:
9750:
9749:
9183:matrix sub-blocks
9155:
9154:
8592:
8469:
8454:
8321:
7781:
7429:
7414:
6762:
6655:
6456:
6328:
6253:
6155:cofactor equation
6067:
6004:
5721:
5676:
5596:Analytic solution
5465:positive definite
5428:
5309:orthogonal matrix
5203:th column is the
4984:
4894:
4555:
4405:
4364:
3428:
3266:
3235:
3026:
2953:
2882:is 0, which is a
2825:
2809:
2751:
2658:
2480:topological space
2463:Furthermore, the
2349:
2291:
2219:
2135:involutory matrix
858:column-equivalent
728:(e.g., the field
390:(multiplicative)
244:
243:
236:
226:
225:
218:
171:
170:
163:
145:
68:
16:(Redirected from
18306:
18258:List of matrices
18250:
18249:
18226:Row echelon form
18170:State transition
18099:Seidel adjacency
17981:Totally positive
17841:Alternating sign
17438:Complex Hadamard
17341:
17334:
17327:
17318:
17317:
17304:
17303:
17186:Exterior algebra
17123:Hadamard product
17040:
17028:Linear equations
16899:
16892:
16885:
16876:
16875:
16866:
16851:
16825:
16818:
16814:
16811:
16805:
16773:
16772:
16765:
16756:
16754:
16744:
16719:
16683:
16656:
16655:
16653:
16629:
16623:
16621:
16597:
16591:
16590:
16572:
16566:
16565:
16563:
16561:
16547:
16541:
16540:
16538:
16506:
16500:
16499:
16481:
16475:
16461:
16455:
16448:
16439:
16438:
16420:
16414:
16413:
16395:
16389:
16388:
16379:(1–2): 119–129.
16368:
16362:
16357:
16337:
16331:
16330:
16289:
16283:
16282:
16271:
16265:
16264:
16253:
16247:
16245:
16223:
16217:
16216:
16214:
16213:
16198:
16192:
16191:
16183:
16177:
16176:
16161:(3rd ed.).
16147:
16090:LU decomposition
15983:LU decomposition
15933:
15931:
15930:
15925:
15923:
15916:
15912:
15911:
15898:
15897:
15888:
15887:
15861:
15855:
15850:
15849:
15841:
15834:
15829:
15808:
15807:
15799:
15786:
15785:
15773:
15762:
15747:
15743:
15742:
15729:
15728:
15719:
15718:
15707:
15701:
15696:
15695:
15684:
15677:
15672:
15651:
15650:
15645:
15632:
15631:
15622:
15611:
15586:
15584:
15583:
15578:
15576:
15569:
15568:
15542:
15536:
15534:
15530:
15524:
15523:
15518:
15512:
15510:
15509:
15501:
15494:
15489:
15464:
15462:
15458:
15452:
15451:
15450:
15442:
15436:
15430:
15421:
15420:
15409:
15403:
15401:
15397:
15391:
15390:
15385:
15379:
15377:
15376:
15365:
15358:
15353:
15331:
15329:
15325:
15319:
15318:
15317:
15312:
15306:
15300:
15284:
15282:
15281:
15276:
15261:
15259:
15258:
15253:
15248:
15244:
15243:
15230:
15229:
15217:
15209:
15208:
15199:
15191:
15183:
15182:
15172:
15154:
15137:
15126:
15102:
15100:
15099:
15094:
15086:
15078:
15077:
15068:
15066:
15062:
15056:
15055:
15050:
15044:
15039:
15031:
15030:
15020:
15008:
15006:
15002:
14996:
14992:
14981:
14975:
14960:
14958:
14957:
14952:
14943:
14942:
14930:
14929:
14920:
14919:
14911:
14905:
14900:
14899:
14891:
14879:
14878:
14870:
14861:
14860:
14852:
14848:
14847:
14836:
14815:
14813:
14812:
14807:
14792:
14790:
14789:
14784:
14779:
14778:
14770:
14764:
14762:
14758:
14752:
14751:
14746:
14740:
14738:
14737:
14729:
14717:
14715:
14711:
14705:
14704:
14703:
14695:
14689:
14683:
14671:
14669:
14668:
14663:
14661:
14660:
14652:
14639:
14637:
14636:
14631:
14629:
14627:
14623:
14617:
14616:
14611:
14605:
14603:
14602:
14594:
14578:
14576:
14575:
14570:
14565:
14557:
14555:
14551:
14545:
14544:
14539:
14533:
14528:
14526:
14522:
14516:
14515:
14510:
14504:
14502:
14501:
14493:
14484:
14479:
14477:
14473:
14467:
14466:
14465:
14457:
14451:
14445:
14440:
14438:
14434:
14428:
14424:
14419:
14418:
14410:
14401:
14395:
14379:
14377:
14376:
14371:
14369:
14361:
14356:
14355:
14347:
14327:
14325:
14324:
14319:
14314:
14313:
14305:
14299:
14297:
14293:
14287:
14286:
14281:
14275:
14273:
14272:
14264:
14252:
14250:
14246:
14240:
14239:
14238:
14230:
14224:
14218:
14203:with respect to
14182:
14180:
14179:
14174:
14172:
14160:
14158:
14157:
14152:
14141:
14140:
14135:
14120:
14119:
14114:
14105:
14104:
14099:
14083:
14081:
14080:
14075:
14073:
14072:
14067:
14054:
14052:
14051:
14046:
14044:
14043:
14038:
14029:
14024:
14019:
14003:
13999:
13998:
13997:
13992:
13983:
13982:
13977:
13963:
13959:
13958:
13954:
13953:
13952:
13947:
13938:
13937:
13932:
13921:
13917:
13916:
13915:
13910:
13901:
13900:
13895:
13876:
13871:
13870:
13862:
13845:
13843:
13842:
13837:
13834:
13829:
13813:
13811:
13810:
13805:
13803:
13802:
13797:
13788:
13783:
13778:
13762:
13758:
13757:
13756:
13751:
13742:
13741:
13736:
13722:
13721:
13713:
13707:
13695:
13693:
13692:
13687:
13685:
13684:
13679:
13666:
13664:
13663:
13658:
13656:
13655:
13650:
13637:
13635:
13634:
13629:
13627:
13626:
13607:
13605:
13604:
13599:
13591:
13590:
13572:
13571:
13563:
13547:
13545:
13544:
13539:
13537:
13536:
13524:
13523:
13518:
13503:
13502:
13497:
13489:
13485:
13484:
13479:
13464:
13463:
13458:
13443:
13442:
13421:
13420:
13415:
13406:
13405:
13378:
13377:
13372:
13366:
13365:
13350:
13349:
13344:
13319:Clifford algebra
13316:
13314:
13313:
13308:
13305:
13300:
13288:
13287:
13282:
13273:
13272:
13267:
13258:
13257:
13252:
13243:
13242:
13237:
13224:
13222:
13221:
13216:
13214:
13213:
13208:
13188:
13186:
13185:
13180:
13178:
13177:
13172:
13155:
13153:
13152:
13147:
13145:
13144:
13139:
13133:
13132:
13117:
13116:
13111:
13098:
13094:
13090:
13088:
13087:
13082:
13052:
13050:
13049:
13044:
13042:
13038:
13037:
13018:
13006:
12993:Reciprocal basis
12982:
12970:
12966:
12962:
12954:
12940:
12923:
12906:
12904:
12903:
12898:
12891:
12890:
12888:
12887:
12883:
12879:
12871:
12863:
12862:
12854:
12830:
12829:
12828:
12820:
12811:
12803:
12795:
12794:
12786:
12779:
12774:
12773:
12765:
12756:
12755:
12747:
12727:
12714:
12712:
12711:
12706:
12699:
12698:
12697:
12689:
12683:
12682:
12677:
12673:
12669:
12661:
12653:
12652:
12644:
12631:
12626:
12608:
12607:
12599:
12583:
12574:
12572:
12571:
12566:
12558:
12557:
12552:
12548:
12547:
12546:
12538:
12532:
12524:
12512:
12497:
12494:
12491:
12482:
12479:
12471:
12470:
12465:
12461:
12460:
12455:
12454:
12446:
12437:
12425:
12400:
12394:
12385:
12381:
12374:Therefore, only
12369:
12367:
12366:
12361:
12359:
12355:
12354:
12353:
12352:
12351:
12337:
12329:
12318:
12307:
12296:
12278:
12277:
12268:
12260:
12251:
12244:
12243:
12233:
12197:
12195:
12194:
12189:
12184:
12183:
12174:
12166:
12157:
12152:
12134:
12133:
12125:
12105:
12096:
12094:
12093:
12088:
12080:
12079:
12070:
12062:
12053:
12028:
12014:
11999:
11973:
11967:
11965:
11964:
11959:
11957:
11952:
11951:
11946:
11930:
11928:
11927:
11922:
11920:
11919:
11914:
11908:
11907:
11899:
11895:
11894:
11889:
11888:
11883:
11868:
11867:
11859:
11846:
11844:
11843:
11838:
11836:
11824:
11822:
11821:
11816:
11814:
11809:
11808:
11803:
11790:
11788:
11787:
11782:
11780:
11768:
11766:
11765:
11760:
11755:
11754:
11746:
11740:
11735:
11734:
11726:
11720:
11719:
11714:
11708:
11707:
11699:
11690:
11689:
11684:
11677:
11672:
11667:
11658:
11657:
11652:
11640:
11638:
11637:
11632:
11627:
11626:
11618:
11612:
11607:
11606:
11598:
11589:
11588:
11583:
11577:
11576:
11568:
11559:
11558:
11553:
11541:
11539:
11538:
11533:
11528:
11527:
11522:
11516:
11515:
11507:
11501:
11493:
11492:
11487:
11481:
11480:
11475:
11466:
11465:
11460:
11448:
11446:
11445:
11440:
11435:
11434:
11426:
11420:
11412:
11411:
11406:
11393:
11387:
11381:
11379:
11378:
11373:
11371:
11370:
11365:
11359:
11358:
11350:
11344:
11336:
11328:
11309:
11300:
11298:
11297:
11292:
11287:
11286:
11279:
11278:
11270:
11262:
11261:
11253:
11247:
11242:
11241:
11233:
11220:
11219:
11211:
11205:
11204:
11199:
11193:
11192:
11184:
11173:
11172:
11164:
11158:
11153:
11152:
11144:
11138:
11137:
11132:
11126:
11125:
11117:
11108:
11107:
11099:
11082:
11081:
11073:
11072:
11065:
11058:
11049:
11048:
11043:
11035:
11013:
11009:
11000:
10998:
10997:
10992:
10987:
10986:
10979:
10978:
10970:
10962:
10961:
10953:
10944:
10943:
10935:
10922:
10915:
10914:
10906:
10889:
10888:
10880:
10879:
10872:
10865:
10856:
10849:
10821:
10815:
10805:
10785:
10776:
10774:
10773:
10768:
10763:
10762:
10755:
10748:
10747:
10739:
10733:
10721:
10720:
10712:
10706:
10696:
10683:
10682:
10675:
10674:
10666:
10662:
10661:
10656:
10655:
10647:
10641:
10633:
10620:
10611:
10604:
10603:
10595:
10591:
10590:
10585:
10584:
10576:
10570:
10562:
10540:
10539:
10531:
10530:
10523:
10516:
10507:
10500:
10478:
10474:
10468:
10446:where Equation (
10438:
10429:
10427:
10426:
10421:
10419:
10415:
10414:
10406:
10402:
10401:
10396:
10395:
10387:
10375:
10357:
10356:
10348:
10339:
10338:
10330:
10326:
10325:
10320:
10319:
10311:
10299:
10288:
10283:
10282:
10274:
10265:
10264:
10256:
10246:
10245:
10237:
10228:
10227:
10219:
10215:
10214:
10209:
10208:
10200:
10188:
10170:
10169:
10161:
10157:
10156:
10151:
10150:
10142:
10130:
10119:
10114:
10113:
10105:
10095:
10094:
10086:
10082:
10081:
10076:
10075:
10067:
10055:
10044:
10039:
10038:
10030:
10017:
10016:
10008:
9999:
9998:
9990:
9986:
9985:
9980:
9979:
9971:
9959:
9944:
9943:
9935:
9926:
9925:
9917:
9913:
9912:
9907:
9906:
9898:
9886:
9875:
9870:
9869:
9861:
9852:
9851:
9843:
9830:
9829:
9821:
9817:
9816:
9811:
9810:
9802:
9790:
9767:
9744:
9735:
9733:
9732:
9727:
9722:
9721:
9714:
9713:
9705:
9696:
9695:
9687:
9683:
9682:
9677:
9676:
9668:
9656:
9645:
9640:
9639:
9631:
9622:
9621:
9613:
9604:
9603:
9595:
9591:
9590:
9585:
9584:
9576:
9564:
9553:
9548:
9547:
9539:
9526:
9525:
9517:
9508:
9507:
9499:
9495:
9494:
9489:
9488:
9480:
9468:
9452:
9451:
9443:
9439:
9438:
9433:
9432:
9424:
9412:
9390:
9389:
9381:
9380:
9373:
9366:
9357:
9350:
9328:
9324:
9311:
9305:
9299:
9293:
9287:
9272:
9266:
9238:
9231:Schur complement
9228:
9216:is diagonal and
9215:
9209:
9196:
9190:
9180:
9174:
9168:
9162:
9149:
9140:
9138:
9137:
9132:
9127:
9126:
9119:
9118:
9110:
9106:
9105:
9100:
9099:
9091:
9079:
9066:
9065:
9057:
9048:
9047:
9039:
9035:
9034:
9029:
9028:
9020:
9008:
8990:
8989:
8981:
8977:
8976:
8971:
8970:
8962:
8950:
8939:
8934:
8933:
8925:
8914:
8913:
8905:
8896:
8895:
8887:
8883:
8882:
8877:
8876:
8868:
8856:
8845:
8840:
8839:
8831:
8822:
8821:
8813:
8796:
8795:
8787:
8786:
8779:
8772:
8763:
8756:
8734:
8718:
8716:
8715:
8710:
8705:
8701:
8700:
8699:
8694:
8685:
8674:
8673:
8668:
8659:
8655:
8651:
8650:
8645:
8627:
8626:
8617:
8598:
8593:
8585:
8580:
8575:
8571:
8567:
8566:
8561:
8537:
8536:
8531:
8516:
8499:
8498:
8489:
8470:
8462:
8455:
8453:
8449:
8434:
8429:
8428:
8420:
8404:
8397:
8383:
8381:
8380:
8375:
8367:
8366:
8361:
8352:
8351:
8346:
8334:
8333:
8332:
8322:
8320:
8316:
8315:
8310:
8301:
8300:
8295:
8283:
8282:
8281:
8267:
8249:
8233:
8231:
8230:
8225:
8220:
8219:
8214:
8205:
8204:
8199:
8187:
8186:
8181:
8169:
8148:
8146:
8145:
8140:
8138:
8133:
8132:
8124:
8115:
8103:
8097:
8087:
8085:
8084:
8079:
8071:
8070:
8065:
8056:
8055:
8050:
8038:
8037:
8032:
8020:
7995:
7986:
7977:
7968:
7960:
7951:
7949:
7948:
7943:
7938:
7937:
7930:
7929:
7928:
7922:
7918:
7917:
7912:
7903:
7902:
7897:
7882:
7881:
7880:
7874:
7870:
7869:
7864:
7855:
7854:
7849:
7834:
7833:
7832:
7826:
7822:
7821:
7816:
7807:
7806:
7801:
7782:
7780:
7776:
7761:
7756:
7755:
7747:
7731:
7729:
7728:
7723:
7721:
7720:
7715:
7702:
7700:
7699:
7694:
7692:
7691:
7686:
7673:
7671:
7670:
7665:
7663:
7662:
7657:
7644:
7642:
7641:
7636:
7634:
7633:
7626:
7625:
7620:
7612:
7611:
7606:
7598:
7597:
7592:
7575:
7555:
7548:
7546:
7545:
7540:
7535:
7531:
7530:
7529:
7524:
7515:
7504:
7496:
7491:
7487:
7483:
7482:
7477:
7459:
7458:
7449:
7430:
7422:
7415:
7413:
7409:
7394:
7389:
7388:
7380:
7361:
7359:
7358:
7353:
7318:
7293:
7284:
7282:
7281:
7276:
7274:
7244:
7198:
7155:
7110:
7067:
7021:
6979:
6933:
6890:
6861:
6855:
6845:
6843:
6842:
6837:
6835:
6834:
6763:
6761:
6757:
6742:
6737:
6736:
6735:
6729:
6728:
6656:
6654:
6650:
6635:
6630:
6629:
6621:
6620:
6554:
6553:
6545:
6529:
6514:
6512:
6511:
6506:
6501:
6497:
6496:
6488:
6483:
6479:
6478:
6457:
6455:
6454:
6442:
6437:
6436:
6428:
6405:
6390:
6388:
6387:
6382:
6377:
6376:
6329:
6327:
6307:
6302:
6301:
6254:
6252:
6251:
6239:
6234:
6233:
6225:
6224:
6185:
6184:
6176:
6160:
6140:
6130:
6124:
6114:
6112:
6101:
6099:
6098:
6093:
6091:
6087:
6086:
6078:
6068:
6066:
6065:
6058:
6042:
6037:
6036:
6028:
6024:
6023:
6022:
6016:
6005:
6003:
6002:
5995:
5979:
5974:
5973:
5965:
5961:
5960:
5952:
5928:
5926:
5925:
5920:
5918:
5917:
5910:
5909:
5901:
5888:
5887:
5879:
5871:
5870:
5862:
5830:
5829:
5821:
5808:
5807:
5802:
5794:
5793:
5788:
5778:
5777:
5769:
5756:
5755:
5750:
5742:
5741:
5736:
5722:
5720:
5719:
5712:
5696:
5691:
5690:
5689:
5683:
5677:
5675:
5674:
5667:
5651:
5646:
5645:
5637:
5591:
5581:
5574:
5564:lower triangular
5561:
5552:
5550:
5549:
5544:
5539:
5538:
5530:
5524:
5523:
5515:
5511:
5510:
5505:
5491:
5490:
5482:
5462:
5442:
5440:
5439:
5434:
5429:
5427:
5426:
5414:
5409:
5408:
5400:
5396:
5395:
5368:
5362:
5360:
5359:
5354:
5349:
5348:
5347:
5341:
5332:
5331:
5323:
5306:
5300:
5294:
5292:
5291:
5286:
5281:
5280:
5268:
5267:
5244:
5238:
5232:
5230:
5229:
5224:
5222:
5221:
5202:
5198:
5186:
5177:
5175:
5174:
5169:
5164:
5163:
5155:
5149:
5148:
5140:
5134:
5126:
5125:
5117:
5101:
5095:
5068:
5066:
5065:
5060:
5052:
5051:
5027:
5026:
5014:
5013:
5001:
5000:
4985:
4983:
4963:
4962:
4961:
4936:
4934:
4933:
4922:
4915:
4910:
4895:
4893:
4889:
4874:
4869:
4868:
4860:
4844:
4842:
4841:
4836:
4834:
4830:
4829:
4786:
4785:
4768:Bell polynomials
4762:
4760:
4759:
4754:
4740:
4739:
4726:
4715:
4681:
4679:
4678:
4673:
4665:
4664:
4648:
4640:
4630:
4622:
4616:
4609:
4607:
4606:
4601:
4596:
4595:
4594:
4593:
4583:
4579:
4578:
4573:
4556:
4554:
4550:
4549:
4540:
4539:
4538:
4537:
4522:
4521:
4520:
4513:
4512:
4488:
4485:
4474:
4458:
4457:
4456:
4432:
4431:
4419:
4418:
4404:
4403:
4398:
4391:
4380:
4365:
4363:
4359:
4344:
4339:
4338:
4330:
4314:
4308:
4300:
4252:
4250:
4249:
4244:
4239:
4238:
4226:
4225:
4213:
4212:
4197:
4196:
4155:
4149:
4139:
4133:
4127:
4125:
4124:
4119:
4114:
4109:
4108:
4103:
4097:
4096:
4091:
4082:
4081:
4070:
4064:
4063:
4058:
4042:
4040:
4039:
4034:
4029:
4028:
4020:
4011:
4010:
4002:
3996:
3984:
3982:
3981:
3976:
3971:
3970:
3962:
3956:
3948:
3943:
3942:
3937:
3931:
3930:
3925:
3916:
3915:
3904:
3898:
3897:
3892:
3879:
3877:
3876:
3871:
3866:
3865:
3857:
3842:
3840:
3839:
3834:
3829:
3821:
3816:
3815:
3810:
3804:
3803:
3798:
3789:
3788:
3777:
3771:
3770:
3765:
3752:
3750:
3749:
3744:
3742:
3741:
3736:
3717:
3715:
3714:
3709:
3704:
3703:
3665:
3664:
3656:
3641:
3639:
3638:
3633:
3628:
3624:
3565:
3563:
3562:
3557:
3549:
3548:
3536:
3535:
3512:
3510:
3509:
3504:
3499:
3498:
3486:
3485:
3461:
3459:
3458:
3453:
3448:
3444:
3430:
3421:
3367:
3365:
3364:
3359:
3351:
3350:
3338:
3337:
3321:
3320:
3299:
3297:
3296:
3291:
3286:
3282:
3268:
3259:
3237:
3228:
3202:
3200:
3199:
3194:
3186:
3185:
3173:
3172:
3160:
3159:
3140:
3138:
3137:
3132:
3130:
3129:
3113:
3111:
3110:
3105:
3103:
3102:
3084:
3082:
3081:
3076:
3071:
3067:
3028:
3019:
2991:
2989:
2988:
2983:
2978:
2977:
2955:
2946:
2924:
2904:augmented matrix
2881:
2879:
2878:
2873:
2871:
2856:
2854:
2853:
2848:
2843:
2842:
2827:
2818:
2811:
2802:
2780:
2762:
2760:
2759:
2754:
2752:
2744:
2736:
2721:
2719:
2718:
2713:
2711:
2696:
2694:
2693:
2688:
2683:
2682:
2660:
2651:
2629:
2611:
2598:
2596:
2595:
2590:
2585:
2584:
2546:
2531:
2505:
2501:
2498:in the space of
2489:
2485:
2470:
2466:
2459:
2455:
2441:because it is a
2421:
2419:
2417:
2416:
2411:
2406:
2405:
2394:
2375:
2371:
2359:
2357:
2356:
2351:
2347:
2346:
2338:
2320:
2314:
2301:
2299:
2298:
2293:
2289:
2288:
2280:
2250:
2248:
2247:
2242:
2234:
2220:
2218:
2214:
2199:
2194:
2193:
2185:
2169:
2160:
2154:
2133:), is called an
2132:
2122:
2112:
2103:
2093:
2083:
2081:
2080:
2075:
2070:
2069:
2051:
2050:
2040:
2039:
2033:
2013:
2011:
2010:
2005:
1972:
1970:
1969:
1964:
1962:
1961:
1945:
1939:
1937:
1936:
1931:
1928:
1927:
1921:
1905:
1899:
1885:
1875:
1869:
1860:
1858:
1857:
1852:
1847:
1846:
1834:
1820:
1819:
1811:
1794:
1792:
1791:
1786:
1780:
1772:
1767:
1760:
1752:
1747:
1737:
1729:
1718:
1711:
1703:
1698:
1689:
1688:
1676:
1675:
1670:
1664:
1663:
1652:
1643:
1642:
1637:
1631:
1630:
1625:
1609:
1605:
1601:
1599:
1598:
1593:
1591:
1590:
1585:
1570:
1569:
1564:
1551:
1549:
1548:
1543:
1538:
1537:
1529:
1523:
1522:
1514:
1505:
1504:
1492:
1474:
1468:
1462:
1458:
1452:
1450:
1449:
1444:
1442:
1441:
1440:
1430:
1429:
1421:
1409:
1408:
1396:
1395:
1394:
1388:
1370:
1360:
1354:
1352:
1351:
1346:
1344:
1343:
1335:
1329:
1328:
1323:
1314:
1313:
1304:
1285:
1281:
1279:
1278:
1273:
1271:
1270:
1262:
1256:
1255:
1240:
1239:
1227:
1208:
1206:
1205:
1200:
1198:
1190:
1189:
1177:
1176:
1168:
1149:
1140:Other properties
1130:
1121:
1115:
1113:
1112:
1107:
1105:
1094:
1082:
1076:
1074:
1073:
1068:
1056:
1038:commutative ring
1035:
1027:
1014:
1011:form a basis of
1010:
1001:
993:
976:
972:
966:
956:
950:
942:
926:
918:
903:
892:
888:
880:
867:
863:
855:
847:
834:
830:
822:
814:
802:
796:
787:
768:
762:
753:
751:
749:
748:
743:
741:
727:
720:
716:
712:
693:
679:
667:
659:
611:
596:
590:
586:
582:
572:
566:
560:
545:
539:
533:
527:
521:
511:
505:
495:
491:
487:
473:
463:
459:
411:Matrix inversion
408:
402:
387:
381:
368:
364:
360:
349:
347:
346:
341:
336:
335:
330:
321:
310:
295:
289:
285:
265:
257:
253:
239:
232:
221:
214:
210:
207:
201:
181:
180:
173:
166:
159:
155:
152:
146:
144:
103:
79:
71:
60:
38:
37:
30:
21:
18:Matrix inversion
18314:
18313:
18309:
18308:
18307:
18305:
18304:
18303:
18274:
18273:
18272:
18267:
18244:
18235:
18184:
18108:
18054:
17990:
17824:
17742:
17688:
17627:
17428:Centrosymmetric
17351:
17345:
17315:
17310:
17292:
17254:
17210:
17147:
17099:
17041:
17032:
16998:Change of basis
16988:Multilinear map
16926:
16908:
16903:
16826:
16815:
16809:
16806:
16787:
16778:This article's
16774:
16770:
16763:
16752:
16737:
16716:
16700:Stein, Clifford
16668:
16665:
16663:Further reading
16660:
16659:
16630:
16626:
16619:
16598:
16594:
16587:
16573:
16569:
16559:
16557:
16555:cs.uwaterloo.ca
16549:
16548:
16544:
16507:
16503:
16496:
16482:
16478:
16462:
16458:
16449:
16442:
16435:
16421:
16417:
16410:
16396:
16392:
16369:
16365:
16354:
16338:
16334:
16290:
16286:
16272:
16268:
16254:
16250:
16242:
16228:Matrix Analysis
16224:
16220:
16211:
16209:
16199:
16195:
16184:
16180:
16173:
16148:
16144:
16139:
16134:
16080:
16042:, are sent via
16020:
16000:
15991:
15964:
15940:
15921:
15920:
15907:
15903:
15899:
15893:
15892:
15862:
15857:
15856:
15851:
15842:
15837:
15836:
15830:
15819:
15800:
15795:
15794:
15787:
15778:
15774:
15769:
15758:
15752:
15751:
15738:
15734:
15730:
15724:
15723:
15708:
15703:
15702:
15697:
15685:
15680:
15679:
15673:
15662:
15646:
15641:
15640:
15633:
15627:
15623:
15618:
15607:
15600:
15598:
15595:
15594:
15574:
15573:
15543:
15538:
15537:
15526:
15525:
15519:
15514:
15513:
15511:
15502:
15497:
15496:
15490:
15479:
15465:
15454:
15453:
15443:
15438:
15437:
15432:
15431:
15429:
15426:
15425:
15410:
15405:
15404:
15393:
15392:
15386:
15381:
15380:
15378:
15366:
15361:
15360:
15354:
15343:
15332:
15321:
15320:
15313:
15308:
15307:
15302:
15301:
15299:
15295:
15293:
15290:
15289:
15270:
15267:
15266:
15239:
15235:
15231:
15225:
15224:
15213:
15204:
15200:
15195:
15187:
15178:
15174:
15168:
15150:
15133:
15122:
15114:
15111:
15110:
15082:
15073:
15069:
15058:
15057:
15051:
15046:
15045:
15043:
15035:
15026:
15022:
15016:
14998:
14997:
14988:
14977:
14976:
14974:
14972:
14969:
14968:
14938:
14934:
14925:
14924:
14912:
14907:
14906:
14901:
14892:
14887:
14886:
14871:
14866:
14865:
14853:
14843:
14832:
14831:
14827:
14826:
14824:
14821:
14820:
14801:
14798:
14797:
14771:
14766:
14765:
14754:
14753:
14747:
14742:
14741:
14739:
14730:
14725:
14724:
14707:
14706:
14696:
14691:
14690:
14685:
14684:
14682:
14680:
14677:
14676:
14653:
14648:
14647:
14645:
14642:
14641:
14619:
14618:
14612:
14607:
14606:
14604:
14595:
14590:
14589:
14587:
14584:
14583:
14561:
14547:
14546:
14540:
14535:
14534:
14532:
14518:
14517:
14511:
14506:
14505:
14503:
14494:
14489:
14488:
14480:
14469:
14468:
14458:
14453:
14452:
14447:
14446:
14444:
14430:
14429:
14420:
14411:
14406:
14405:
14397:
14396:
14394:
14392:
14389:
14388:
14365:
14357:
14348:
14343:
14342:
14340:
14337:
14336:
14306:
14301:
14300:
14289:
14288:
14282:
14277:
14276:
14274:
14265:
14260:
14259:
14242:
14241:
14231:
14226:
14225:
14220:
14219:
14217:
14215:
14212:
14211:
14189:
14168:
14166:
14163:
14162:
14161:and the matrix
14136:
14131:
14130:
14115:
14110:
14109:
14100:
14095:
14094:
14089:
14086:
14085:
14068:
14063:
14062:
14060:
14057:
14056:
14039:
14034:
14033:
14020:
14015:
14007:
13993:
13988:
13987:
13978:
13973:
13972:
13971:
13967:
13948:
13943:
13942:
13933:
13928:
13927:
13926:
13922:
13911:
13906:
13905:
13896:
13891:
13890:
13889:
13885:
13884:
13880:
13872:
13863:
13858:
13857:
13855:
13852:
13851:
13850:. We also have
13848:Kronecker delta
13830:
13825:
13819:
13816:
13815:
13798:
13793:
13792:
13779:
13774:
13766:
13752:
13747:
13746:
13737:
13732:
13731:
13730:
13726:
13714:
13709:
13708:
13703:
13701:
13698:
13697:
13696:. We then have
13680:
13675:
13674:
13672:
13669:
13668:
13651:
13646:
13645:
13643:
13640:
13639:
13622:
13618:
13613:
13610:
13609:
13583:
13579:
13564:
13559:
13558:
13556:
13553:
13552:
13529:
13525:
13519:
13514:
13513:
13498:
13493:
13492:
13480:
13475:
13474:
13459:
13454:
13453:
13438:
13434:
13416:
13411:
13410:
13395:
13391:
13373:
13368:
13367:
13358:
13354:
13345:
13340:
13339:
13337:
13334:
13333:
13301:
13296:
13283:
13278:
13277:
13268:
13263:
13262:
13253:
13248:
13247:
13238:
13233:
13232:
13230:
13227:
13226:
13209:
13204:
13203:
13201:
13198:
13197:
13195:Euclidean space
13189:are a standard
13173:
13168:
13167:
13165:
13162:
13161:
13140:
13135:
13134:
13125:
13121:
13112:
13107:
13106:
13104:
13101:
13100:
13096:
13092:
13058:
13055:
13054:
13030:
13026:
13022:
13014:
13012:
13009:
13008:
12998:
12995:
12989:
12972:
12968:
12964:
12956:
12944:
12938:
12919:
12916:
12875:
12867:
12855:
12850:
12849:
12848:
12844:
12831:
12821:
12816:
12815:
12807:
12799:
12787:
12782:
12781:
12780:
12778:
12766:
12761:
12760:
12748:
12743:
12742:
12740:
12737:
12736:
12719:
12690:
12685:
12684:
12678:
12665:
12657:
12645:
12640:
12639:
12638:
12634:
12633:
12627:
12616:
12600:
12595:
12594:
12592:
12589:
12588:
12579:
12553:
12539:
12534:
12533:
12528:
12520:
12519:
12515:
12514:
12502:
12478:
12466:
12456:
12447:
12442:
12441:
12433:
12432:
12428:
12427:
12415:
12409:
12406:
12405:
12396:
12390:
12383:
12375:
12347:
12343:
12342:
12338:
12333:
12325:
12314:
12313:
12309:
12297:
12286:
12273:
12269:
12264:
12256:
12239:
12235:
12234:
12223:
12217:
12214:
12213:
12179:
12175:
12170:
12162:
12153:
12142:
12126:
12121:
12120:
12118:
12115:
12114:
12101:
12075:
12071:
12066:
12058:
12043:
12037:
12034:
12033:
12024:
12021:
12001:
11990:
11969:
11953:
11947:
11942:
11941:
11939:
11936:
11935:
11915:
11910:
11909:
11900:
11890:
11884:
11879:
11878:
11877:
11873:
11872:
11860:
11855:
11854:
11852:
11849:
11848:
11832:
11830:
11827:
11826:
11810:
11804:
11799:
11798:
11796:
11793:
11792:
11776:
11774:
11771:
11770:
11747:
11742:
11741:
11736:
11727:
11722:
11721:
11715:
11710:
11709:
11700:
11695:
11694:
11685:
11680:
11679:
11673:
11668:
11663:
11653:
11648:
11647:
11645:
11642:
11641:
11619:
11614:
11613:
11608:
11599:
11594:
11593:
11584:
11579:
11578:
11569:
11564:
11563:
11554:
11549:
11548:
11546:
11543:
11542:
11523:
11518:
11517:
11508:
11503:
11502:
11497:
11488:
11483:
11482:
11476:
11471:
11470:
11461:
11456:
11455:
11453:
11450:
11449:
11427:
11422:
11421:
11416:
11407:
11402:
11401:
11399:
11396:
11395:
11389:
11383:
11366:
11361:
11360:
11351:
11346:
11345:
11340:
11332:
11324:
11322:
11319:
11318:
11281:
11280:
11271:
11266:
11265:
11263:
11254:
11249:
11248:
11243:
11234:
11229:
11228:
11222:
11221:
11212:
11207:
11206:
11200:
11195:
11194:
11185:
11180:
11179:
11174:
11165:
11160:
11159:
11154:
11145:
11140:
11139:
11133:
11128:
11127:
11118:
11113:
11112:
11100:
11095:
11094:
11087:
11086:
11074:
11067:
11066:
11061:
11059:
11054:
11051:
11050:
11044:
11039:
11038:
11036:
11031:
11024:
11023:
11022:
11020:
11017:
11016:
11005:
10981:
10980:
10971:
10966:
10965:
10963:
10954:
10946:
10945:
10936:
10931:
10930:
10924:
10923:
10918:
10916:
10907:
10902:
10901:
10894:
10893:
10881:
10874:
10873:
10868:
10866:
10861:
10858:
10857:
10852:
10850:
10845:
10838:
10837:
10836:
10834:
10831:
10830:
10824:pseudo inverses
10817:
10811:
10801:
10757:
10756:
10751:
10749:
10740:
10735:
10734:
10729:
10723:
10722:
10713:
10708:
10707:
10702:
10697:
10692:
10685:
10684:
10677:
10676:
10667:
10657:
10648:
10643:
10642:
10637:
10629:
10628:
10624:
10623:
10621:
10616:
10613:
10612:
10607:
10605:
10596:
10586:
10577:
10572:
10571:
10566:
10558:
10557:
10553:
10552:
10545:
10544:
10532:
10525:
10524:
10519:
10517:
10512:
10509:
10508:
10503:
10501:
10496:
10489:
10488:
10487:
10485:
10482:
10481:
10470:
10464:
10417:
10416:
10407:
10397:
10388:
10380:
10379:
10371:
10370:
10366:
10365:
10358:
10349:
10341:
10340:
10331:
10321:
10312:
10304:
10303:
10295:
10294:
10290:
10289:
10284:
10275:
10270:
10269:
10257:
10252:
10251:
10248:
10247:
10238:
10230:
10229:
10220:
10210:
10201:
10193:
10192:
10184:
10183:
10179:
10178:
10171:
10162:
10152:
10143:
10135:
10134:
10126:
10125:
10121:
10120:
10115:
10106:
10101:
10100:
10097:
10096:
10087:
10077:
10068:
10060:
10059:
10051:
10050:
10046:
10045:
10040:
10031:
10026:
10025:
10018:
10009:
10001:
10000:
9991:
9981:
9972:
9964:
9963:
9955:
9954:
9950:
9949:
9946:
9945:
9936:
9928:
9927:
9918:
9908:
9899:
9891:
9890:
9882:
9881:
9877:
9876:
9871:
9862:
9857:
9856:
9844:
9839:
9838:
9831:
9822:
9812:
9803:
9795:
9794:
9786:
9785:
9781:
9780:
9776:
9774:
9771:
9770:
9716:
9715:
9706:
9698:
9697:
9688:
9678:
9669:
9661:
9660:
9652:
9651:
9647:
9646:
9641:
9632:
9627:
9626:
9614:
9609:
9608:
9605:
9596:
9586:
9577:
9569:
9568:
9560:
9559:
9555:
9554:
9549:
9540:
9535:
9534:
9528:
9527:
9518:
9510:
9509:
9500:
9490:
9481:
9473:
9472:
9464:
9463:
9459:
9458:
9453:
9444:
9434:
9425:
9417:
9416:
9408:
9407:
9403:
9402:
9395:
9394:
9382:
9375:
9374:
9369:
9367:
9362:
9359:
9358:
9353:
9351:
9346:
9339:
9338:
9337:
9335:
9332:
9331:
9313:
9307:
9301:
9295:
9289:
9283:
9268:
9262:
9259:nullity theorem
9234:
9217:
9211:
9198:
9192:
9186:
9176:
9170:
9164:
9158:
9121:
9120:
9111:
9101:
9092:
9084:
9083:
9075:
9074:
9070:
9069:
9067:
9058:
9050:
9049:
9040:
9030:
9021:
9013:
9012:
9004:
9003:
8999:
8998:
8992:
8991:
8982:
8972:
8963:
8955:
8954:
8946:
8945:
8941:
8940:
8935:
8926:
8921:
8920:
8915:
8906:
8898:
8897:
8888:
8878:
8869:
8861:
8860:
8852:
8851:
8847:
8846:
8841:
8832:
8827:
8826:
8814:
8809:
8808:
8801:
8800:
8788:
8781:
8780:
8775:
8773:
8768:
8765:
8764:
8759:
8757:
8752:
8745:
8744:
8743:
8741:
8738:
8737:
8725:
8695:
8690:
8689:
8681:
8669:
8664:
8663:
8646:
8641:
8640:
8622:
8618:
8613:
8603:
8599:
8594:
8584:
8576:
8562:
8557:
8556:
8532:
8527:
8526:
8512:
8494:
8490:
8485:
8475:
8471:
8461:
8460:
8456:
8445:
8438:
8433:
8421:
8416:
8415:
8413:
8410:
8409:
8399:
8393:
8390:
8362:
8357:
8356:
8347:
8342:
8341:
8328:
8324:
8323:
8311:
8306:
8305:
8296:
8291:
8290:
8277:
8273:
8272:
8271:
8266:
8258:
8255:
8254:
8238:
8215:
8210:
8209:
8200:
8195:
8194:
8182:
8177:
8176:
8165:
8157:
8154:
8153:
8134:
8125:
8120:
8119:
8111:
8109:
8106:
8105:
8099:
8093:
8066:
8061:
8060:
8051:
8046:
8045:
8033:
8028:
8027:
8016:
8008:
8005:
8004:
7994:
7988:
7985:
7979:
7976:
7970:
7962:
7956:
7932:
7931:
7924:
7923:
7913:
7908:
7907:
7898:
7893:
7892:
7888:
7887:
7884:
7883:
7876:
7875:
7865:
7860:
7859:
7850:
7845:
7844:
7840:
7839:
7836:
7835:
7828:
7827:
7817:
7812:
7811:
7802:
7797:
7796:
7792:
7791:
7784:
7783:
7772:
7765:
7760:
7748:
7743:
7742:
7740:
7737:
7736:
7716:
7711:
7710:
7708:
7705:
7704:
7687:
7682:
7681:
7679:
7676:
7675:
7658:
7653:
7652:
7650:
7647:
7646:
7628:
7627:
7621:
7616:
7615:
7613:
7607:
7602:
7601:
7599:
7593:
7588:
7587:
7580:
7579:
7571:
7569:
7566:
7565:
7553:
7525:
7520:
7519:
7511:
7500:
7492:
7478:
7473:
7472:
7454:
7450:
7445:
7435:
7431:
7421:
7420:
7416:
7405:
7398:
7393:
7381:
7376:
7375:
7373:
7370:
7369:
7314:
7306:
7303:
7302:
7289:
7272:
7271:
7245:
7243:
7236:
7231:
7228:
7199:
7197:
7190:
7185:
7182:
7156:
7154:
7147:
7141:
7140:
7111:
7109:
7102:
7097:
7094:
7068:
7066:
7059:
7054:
7051:
7022:
7020:
7013:
7007:
7006:
6980:
6978:
6971:
6966:
6963:
6934:
6932:
6925:
6920:
6917:
6891:
6889:
6882:
6875:
6873:
6870:
6869:
6857:
6853:
6829:
6828:
6822:
6816:
6809:
6808:
6802:
6796:
6789:
6788:
6782:
6776:
6765:
6764:
6753:
6746:
6741:
6731:
6730:
6723:
6722:
6716:
6710:
6703:
6702:
6696:
6690:
6683:
6682:
6676:
6670:
6659:
6658:
6657:
6646:
6639:
6634:
6622:
6615:
6614:
6609:
6604:
6598:
6597:
6592:
6587:
6581:
6580:
6575:
6570:
6560:
6559:
6558:
6546:
6541:
6540:
6538:
6535:
6534:
6527:
6521:
6492:
6484:
6474:
6467:
6463:
6462:
6458:
6450:
6446:
6441:
6429:
6424:
6423:
6421:
6418:
6417:
6395:
6371:
6370:
6364:
6355:
6354:
6344:
6331:
6330:
6311:
6306:
6296:
6295:
6289:
6280:
6279:
6269:
6256:
6255:
6247:
6243:
6238:
6226:
6219:
6218:
6213:
6207:
6206:
6201:
6191:
6190:
6189:
6177:
6172:
6171:
6169:
6166:
6165:
6158:
6151:
6136:
6126:
6120:
6108:
6106:
6079:
6074:
6073:
6069:
6060:
6059:
6054:
6047:
6046:
6041:
6029:
6018:
6017:
6012:
6011:
6007:
6006:
5997:
5996:
5991:
5984:
5983:
5978:
5966:
5953:
5948:
5947:
5943:
5942:
5940:
5937:
5936:
5912:
5911:
5902:
5897:
5896:
5894:
5889:
5880:
5875:
5874:
5872:
5863:
5858:
5857:
5854:
5853:
5848:
5843:
5838:
5832:
5831:
5822:
5817:
5816:
5814:
5809:
5803:
5798:
5797:
5795:
5789:
5784:
5783:
5780:
5779:
5770:
5765:
5764:
5762:
5757:
5751:
5746:
5745:
5743:
5737:
5732:
5731:
5724:
5723:
5714:
5713:
5708:
5701:
5700:
5695:
5685:
5684:
5679:
5678:
5669:
5668:
5663:
5656:
5655:
5650:
5638:
5633:
5632:
5630:
5627:
5626:
5612:adjugate matrix
5604:
5598:
5587:
5576:
5570:
5557:
5531:
5526:
5525:
5516:
5506:
5501:
5500:
5496:
5495:
5483:
5478:
5477:
5475:
5472:
5471:
5458:
5455:
5449:
5422:
5418:
5413:
5401:
5388:
5384:
5380:
5379:
5377:
5374:
5373:
5364:
5343:
5342:
5337:
5336:
5324:
5319:
5318:
5316:
5313:
5312:
5302:
5296:
5276:
5272:
5260:
5256:
5254:
5251:
5250:
5247:diagonal matrix
5240:
5234:
5217:
5213:
5211:
5208:
5207:
5200:
5188:
5182:
5156:
5151:
5150:
5141:
5136:
5135:
5130:
5118:
5113:
5112:
5110:
5107:
5106:
5097:
5091:
5088:
5082:
5041:
5037:
5022:
5018:
5009:
5005:
4990:
4986:
4964:
4951:
4947:
4937:
4935:
4923:
4918:
4917:
4911:
4900:
4885:
4878:
4873:
4861:
4856:
4855:
4853:
4850:
4849:
4825:
4821:
4817:
4781:
4777:
4775:
4772:
4771:
4735:
4731:
4716:
4705:
4693:
4690:
4689:
4660:
4656:
4654:
4651:
4650:
4646:
4636:
4624:
4618:
4614:
4589:
4585:
4584:
4574:
4569:
4568:
4564:
4563:
4545:
4541:
4533:
4529:
4528:
4524:
4523:
4508:
4504:
4503:
4499:
4489:
4487:
4475:
4464:
4446:
4442:
4427:
4423:
4414:
4410:
4409:
4399:
4394:
4393:
4381:
4370:
4355:
4348:
4343:
4331:
4326:
4325:
4323:
4320:
4319:
4310:
4302:
4296:
4289:
4234:
4230:
4221:
4217:
4208:
4204:
4186:
4182:
4180:
4177:
4176:
4166:Newton's method
4162:
4160:Newton's method
4151:
4145:
4135:
4129:
4110:
4104:
4099:
4098:
4092:
4087:
4086:
4071:
4066:
4065:
4059:
4054:
4053:
4051:
4048:
4047:
4021:
4016:
4015:
4003:
3998:
3997:
3992:
3990:
3987:
3986:
3963:
3958:
3957:
3952:
3944:
3938:
3933:
3932:
3926:
3921:
3920:
3905:
3900:
3899:
3893:
3888:
3887:
3885:
3882:
3881:
3858:
3853:
3852:
3850:
3847:
3846:
3825:
3817:
3811:
3806:
3805:
3799:
3794:
3793:
3778:
3773:
3772:
3766:
3761:
3760:
3758:
3755:
3754:
3737:
3732:
3731:
3729:
3726:
3725:
3698:
3697:
3692:
3686:
3685:
3680:
3670:
3669:
3657:
3652:
3651:
3649:
3646:
3645:
3622:
3621:
3616:
3611:
3606:
3600:
3599:
3594:
3589:
3584:
3577:
3573:
3571:
3568:
3567:
3544:
3540:
3531:
3527:
3518:
3515:
3514:
3513:and row 2 by 2
3494:
3490:
3481:
3477:
3469:
3466:
3465:
3442:
3441:
3436:
3431:
3419:
3417:
3411:
3410:
3402:
3394:
3389:
3379:
3375:
3373:
3370:
3369:
3346:
3342:
3333:
3329:
3316:
3312:
3307:
3304:
3303:
3280:
3279:
3274:
3269:
3257:
3255:
3249:
3248:
3243:
3238:
3226:
3224:
3214:
3210:
3208:
3205:
3204:
3181:
3177:
3168:
3164:
3155:
3151:
3146:
3143:
3142:
3125:
3121:
3119:
3116:
3115:
3098:
3094:
3092:
3089:
3088:
3065:
3064:
3059:
3054:
3046:
3040:
3039:
3034:
3029:
3017:
3015:
3005:
3001:
2999:
2996:
2995:
2972:
2971:
2963:
2957:
2956:
2944:
2942:
2929:
2928:
2920:
2918:
2915:
2914:
2908:identity matrix
2897:
2892:
2867:
2865:
2862:
2861:
2837:
2836:
2828:
2816:
2813:
2812:
2800:
2798:
2785:
2784:
2776:
2774:
2771:
2770:
2743:
2732:
2727:
2724:
2723:
2707:
2705:
2702:
2701:
2677:
2676:
2668:
2662:
2661:
2649:
2647:
2634:
2633:
2625:
2623:
2620:
2619:
2603:
2579:
2578:
2573:
2567:
2566:
2561:
2551:
2550:
2542:
2540:
2537:
2536:
2526:
2523:
2515:ill-conditioned
2503:
2499:
2487:
2483:
2468:
2464:
2457:
2453:
2426:, that is, has
2395:
2390:
2389:
2387:
2384:
2383:
2381:
2373:
2369:
2366:
2342:
2331:
2329:
2326:
2325:
2316:
2310:
2284:
2273:
2271:
2268:
2267:
2257:
2230:
2210:
2203:
2198:
2186:
2181:
2180:
2178:
2175:
2174:
2165:
2156:
2150:
2143:
2124:
2114:
2108:
2099:
2089:
2059:
2055:
2046:
2042:
2035:
2034:
2029:
2023:
2020:
2019:
1978:
1975:
1974:
1957:
1953:
1951:
1948:
1947:
1941:
1923:
1922:
1917:
1911:
1908:
1907:
1906:are denoted as
1901:
1887:
1881:
1871:
1865:
1839:
1835:
1830:
1812:
1807:
1806:
1801:
1798:
1797:
1773:
1768:
1763:
1753:
1748:
1743:
1730:
1719:
1714:
1704:
1699:
1694:
1681:
1677:
1671:
1666:
1665:
1653:
1648:
1647:
1638:
1633:
1632:
1626:
1621:
1620:
1615:
1612:
1611:
1610:matrices, then
1607:
1603:
1602:are invertible
1586:
1581:
1580:
1565:
1560:
1559:
1557:
1554:
1553:
1530:
1525:
1524:
1515:
1510:
1509:
1497:
1493:
1485:
1480:
1477:
1476:
1470:
1464:
1460:
1456:
1436:
1435:
1431:
1422:
1417:
1416:
1401:
1397:
1390:
1389:
1384:
1383:
1378:
1375:
1374:
1366:
1356:
1336:
1331:
1330:
1324:
1319:
1318:
1309:
1305:
1297:
1292:
1289:
1288:
1283:
1263:
1258:
1257:
1248:
1244:
1232:
1228:
1223:
1215:
1212:
1211:
1194:
1182:
1178:
1169:
1164:
1163:
1158:
1155:
1154:
1145:
1142:
1126:
1117:
1101:
1090:
1088:
1085:
1084:
1078:
1062:
1059:
1058:
1052:
1029:
1023:
1012:
1006:
999:
989:
988:The columns of
974:
968:
958:
952:
946:
932:
922:
909:
899:
894:pivot positions
890:
884:
879:
871:
869:identity matrix
865:
861:
851:
846:
838:
836:identity matrix
832:
828:
818:
810:
798:
792:
782:
770:
764:
758:
737:
735:
732:
731:
729:
725:
718:
714:
708:
705:
700:
687:
681:
677:
665:
651:
610:
598:
592:
588:
584:
574:
568:
562:
559:
547:
541:
535:
529:
523:
513:
507:
501:
493:
489:
483:
465:
461:
457:
404:
398:
383:
377:
370:identity matrix
366:
362:
359:
351:
331:
326:
325:
314:
303:
301:
298:
297:
291:
287:
283:
261:
255:
251:
240:
229:
228:
227:
222:
211:
205:
202:
194:help improve it
191:
182:
178:
167:
156:
150:
147:
104:
102:
92:
80:
39:
35:
28:
23:
22:
15:
12:
11:
5:
18312:
18302:
18301:
18296:
18291:
18286:
18284:Linear algebra
18269:
18268:
18266:
18265:
18260:
18255:
18240:
18237:
18236:
18234:
18233:
18228:
18223:
18218:
18216:Perfect matrix
18213:
18208:
18203:
18198:
18192:
18190:
18186:
18185:
18183:
18182:
18177:
18172:
18167:
18162:
18157:
18152:
18147:
18142:
18137:
18132:
18127:
18122:
18116:
18114:
18110:
18109:
18107:
18106:
18101:
18096:
18091:
18086:
18081:
18076:
18071:
18065:
18063:
18056:
18055:
18053:
18052:
18047:
18042:
18037:
18032:
18027:
18022:
18017:
18012:
18007:
18001:
17999:
17992:
17991:
17989:
17988:
17986:Transformation
17983:
17978:
17973:
17968:
17963:
17958:
17953:
17948:
17943:
17938:
17933:
17928:
17923:
17918:
17913:
17908:
17903:
17898:
17893:
17888:
17883:
17878:
17873:
17868:
17863:
17858:
17853:
17848:
17843:
17838:
17832:
17830:
17826:
17825:
17823:
17822:
17817:
17812:
17807:
17802:
17797:
17792:
17787:
17782:
17777:
17772:
17763:
17757:
17755:
17744:
17743:
17741:
17740:
17735:
17730:
17725:
17723:Diagonalizable
17720:
17715:
17710:
17705:
17699:
17697:
17693:Conditions on
17690:
17689:
17687:
17686:
17681:
17676:
17671:
17666:
17661:
17656:
17651:
17646:
17641:
17635:
17633:
17629:
17628:
17626:
17625:
17620:
17615:
17610:
17605:
17600:
17595:
17590:
17585:
17580:
17575:
17573:Skew-symmetric
17570:
17568:Skew-Hermitian
17565:
17560:
17555:
17550:
17545:
17540:
17535:
17530:
17525:
17520:
17515:
17510:
17505:
17500:
17495:
17490:
17485:
17480:
17475:
17470:
17465:
17460:
17455:
17450:
17445:
17440:
17435:
17430:
17425:
17420:
17415:
17410:
17405:
17403:Block-diagonal
17400:
17395:
17390:
17385:
17380:
17378:Anti-symmetric
17375:
17373:Anti-Hermitian
17370:
17365:
17359:
17357:
17353:
17352:
17344:
17343:
17336:
17329:
17321:
17312:
17311:
17309:
17308:
17297:
17294:
17293:
17291:
17290:
17285:
17280:
17275:
17270:
17268:Floating-point
17264:
17262:
17256:
17255:
17253:
17252:
17250:Tensor product
17247:
17242:
17237:
17235:Function space
17232:
17227:
17221:
17219:
17212:
17211:
17209:
17208:
17203:
17198:
17193:
17188:
17183:
17178:
17173:
17171:Triple product
17168:
17163:
17157:
17155:
17149:
17148:
17146:
17145:
17140:
17135:
17130:
17125:
17120:
17115:
17109:
17107:
17101:
17100:
17098:
17097:
17092:
17087:
17085:Transformation
17082:
17077:
17075:Multiplication
17072:
17067:
17062:
17057:
17051:
17049:
17043:
17042:
17035:
17033:
17031:
17030:
17025:
17020:
17015:
17010:
17005:
17000:
16995:
16990:
16985:
16980:
16975:
16970:
16965:
16960:
16955:
16950:
16945:
16940:
16934:
16932:
16931:Basic concepts
16928:
16927:
16925:
16924:
16919:
16913:
16910:
16909:
16906:Linear algebra
16902:
16901:
16894:
16887:
16879:
16873:
16872:
16867:
16852:
16828:
16827:
16782:external links
16777:
16775:
16768:
16762:
16761:External links
16759:
16758:
16757:
16745:
16736:978-0691140391
16735:
16720:
16714:
16684:
16664:
16661:
16658:
16657:
16644:(3): 755–777.
16624:
16617:
16601:Roman, Stephen
16592:
16585:
16567:
16542:
16521:(1): 320–322.
16501:
16494:
16476:
16456:
16440:
16433:
16415:
16408:
16390:
16363:
16352:
16332:
16284:
16266:
16248:
16240:
16234:. p. 14.
16218:
16193:
16178:
16171:
16151:Axler, Sheldon
16141:
16140:
16138:
16135:
16133:
16132:
16127:
16122:
16117:
16112:
16107:
16102:
16097:
16092:
16087:
16081:
16079:
16076:
16040:frequency band
16019:
16016:
15999:
15996:
15990:
15987:
15963:
15960:
15939:
15936:
15935:
15934:
15919:
15915:
15910:
15906:
15902:
15896:
15891:
15886:
15883:
15880:
15877:
15874:
15871:
15868:
15865:
15860:
15854:
15848:
15845:
15840:
15833:
15828:
15825:
15822:
15818:
15814:
15811:
15806:
15803:
15798:
15793:
15790:
15788:
15784:
15781:
15777:
15772:
15768:
15765:
15761:
15757:
15754:
15753:
15750:
15746:
15741:
15737:
15733:
15727:
15722:
15717:
15714:
15711:
15706:
15700:
15694:
15691:
15688:
15683:
15676:
15671:
15668:
15665:
15661:
15657:
15654:
15649:
15644:
15639:
15636:
15634:
15630:
15626:
15621:
15617:
15614:
15610:
15606:
15603:
15602:
15588:
15587:
15572:
15567:
15564:
15561:
15558:
15555:
15552:
15549:
15546:
15541:
15533:
15529:
15522:
15517:
15508:
15505:
15500:
15493:
15488:
15485:
15482:
15478:
15474:
15471:
15468:
15466:
15461:
15457:
15449:
15446:
15441:
15435:
15428:
15427:
15424:
15419:
15416:
15413:
15408:
15400:
15396:
15389:
15384:
15375:
15372:
15369:
15364:
15357:
15352:
15349:
15346:
15342:
15338:
15335:
15333:
15328:
15324:
15316:
15311:
15305:
15298:
15297:
15274:
15263:
15262:
15251:
15247:
15242:
15238:
15234:
15228:
15223:
15220:
15216:
15212:
15207:
15203:
15198:
15194:
15190:
15186:
15181:
15177:
15171:
15167:
15163:
15160:
15157:
15153:
15149:
15146:
15143:
15140:
15136:
15132:
15129:
15125:
15121:
15118:
15104:
15103:
15092:
15089:
15085:
15081:
15076:
15072:
15065:
15061:
15054:
15049:
15042:
15038:
15034:
15029:
15025:
15019:
15015:
15011:
15005:
15001:
14995:
14991:
14987:
14984:
14980:
14962:
14961:
14950:
14946:
14941:
14937:
14933:
14928:
14923:
14918:
14915:
14910:
14904:
14898:
14895:
14890:
14885:
14882:
14877:
14874:
14869:
14864:
14859:
14856:
14851:
14846:
14842:
14839:
14835:
14830:
14805:
14796:Similarly, if
14794:
14793:
14782:
14777:
14774:
14769:
14761:
14757:
14750:
14745:
14736:
14733:
14728:
14723:
14720:
14714:
14710:
14702:
14699:
14694:
14688:
14659:
14656:
14651:
14626:
14622:
14615:
14610:
14601:
14598:
14593:
14580:
14579:
14568:
14564:
14560:
14554:
14550:
14543:
14538:
14531:
14525:
14521:
14514:
14509:
14500:
14497:
14492:
14487:
14483:
14476:
14472:
14464:
14461:
14456:
14450:
14443:
14437:
14433:
14427:
14423:
14417:
14414:
14409:
14404:
14400:
14368:
14364:
14360:
14354:
14351:
14346:
14329:
14328:
14317:
14312:
14309:
14304:
14296:
14292:
14285:
14280:
14271:
14268:
14263:
14258:
14255:
14249:
14245:
14237:
14234:
14229:
14223:
14188:
14185:
14171:
14150:
14147:
14144:
14139:
14134:
14129:
14126:
14123:
14118:
14113:
14108:
14103:
14098:
14093:
14071:
14066:
14042:
14037:
14032:
14028:
14023:
14018:
14014:
14010:
14006:
14002:
13996:
13991:
13986:
13981:
13976:
13970:
13966:
13962:
13957:
13951:
13946:
13941:
13936:
13931:
13925:
13920:
13914:
13909:
13904:
13899:
13894:
13888:
13883:
13879:
13875:
13869:
13866:
13861:
13833:
13828:
13824:
13801:
13796:
13791:
13787:
13782:
13777:
13773:
13769:
13765:
13761:
13755:
13750:
13745:
13740:
13735:
13729:
13725:
13720:
13717:
13712:
13706:
13683:
13678:
13654:
13649:
13625:
13621:
13617:
13597:
13594:
13589:
13586:
13582:
13578:
13575:
13570:
13567:
13562:
13549:
13548:
13535:
13532:
13528:
13522:
13517:
13512:
13509:
13506:
13501:
13496:
13488:
13483:
13478:
13473:
13470:
13467:
13462:
13457:
13452:
13449:
13446:
13441:
13437:
13433:
13430:
13427:
13424:
13419:
13414:
13409:
13404:
13401:
13398:
13394:
13390:
13387:
13384:
13381:
13376:
13371:
13364:
13361:
13357:
13353:
13348:
13343:
13317:), then using
13304:
13299:
13295:
13291:
13286:
13281:
13276:
13271:
13266:
13261:
13256:
13251:
13246:
13241:
13236:
13212:
13207:
13176:
13171:
13143:
13138:
13131:
13128:
13124:
13120:
13115:
13110:
13080:
13077:
13074:
13071:
13068:
13065:
13062:
13041:
13036:
13033:
13029:
13025:
13021:
13017:
13007:square matrix
12991:Main article:
12988:
12985:
12915:
12909:
12908:
12907:
12896:
12886:
12882:
12878:
12874:
12870:
12866:
12861:
12858:
12853:
12847:
12843:
12840:
12837:
12834:
12827:
12824:
12819:
12814:
12810:
12806:
12802:
12798:
12793:
12790:
12785:
12777:
12772:
12769:
12764:
12759:
12754:
12751:
12746:
12716:
12715:
12704:
12696:
12693:
12688:
12681:
12676:
12672:
12668:
12664:
12660:
12656:
12651:
12648:
12643:
12637:
12630:
12625:
12622:
12619:
12615:
12611:
12606:
12603:
12598:
12576:
12575:
12564:
12561:
12556:
12551:
12545:
12542:
12537:
12531:
12527:
12523:
12518:
12511:
12508:
12505:
12501:
12490:
12487:
12477:
12474:
12469:
12464:
12459:
12453:
12450:
12445:
12440:
12436:
12431:
12424:
12421:
12418:
12414:
12372:
12371:
12358:
12350:
12346:
12341:
12336:
12332:
12328:
12324:
12321:
12317:
12312:
12306:
12303:
12300:
12295:
12292:
12289:
12285:
12281:
12276:
12272:
12267:
12263:
12259:
12255:
12250:
12247:
12242:
12238:
12232:
12229:
12226:
12222:
12203:preconditioner
12199:
12198:
12187:
12182:
12178:
12173:
12169:
12165:
12161:
12156:
12151:
12148:
12145:
12141:
12137:
12132:
12129:
12124:
12108:Neumann series
12098:
12097:
12086:
12083:
12078:
12074:
12069:
12065:
12061:
12057:
12052:
12049:
12046:
12042:
12020:
12017:
11956:
11950:
11945:
11918:
11913:
11906:
11903:
11898:
11893:
11887:
11882:
11876:
11871:
11866:
11863:
11858:
11835:
11813:
11807:
11802:
11779:
11758:
11753:
11750:
11745:
11739:
11733:
11730:
11725:
11718:
11713:
11706:
11703:
11698:
11693:
11688:
11683:
11676:
11671:
11666:
11661:
11656:
11651:
11630:
11625:
11622:
11617:
11611:
11605:
11602:
11597:
11592:
11587:
11582:
11575:
11572:
11567:
11562:
11557:
11552:
11531:
11526:
11521:
11514:
11511:
11506:
11500:
11496:
11491:
11486:
11479:
11474:
11469:
11464:
11459:
11438:
11433:
11430:
11425:
11419:
11415:
11410:
11405:
11369:
11364:
11357:
11354:
11349:
11343:
11339:
11335:
11331:
11327:
11313:
11312:
11303:
11301:
11290:
11285:
11277:
11274:
11269:
11264:
11260:
11257:
11252:
11246:
11240:
11237:
11232:
11227:
11224:
11223:
11218:
11215:
11210:
11203:
11198:
11191:
11188:
11183:
11178:
11175:
11171:
11168:
11163:
11157:
11151:
11148:
11143:
11136:
11131:
11124:
11121:
11116:
11111:
11106:
11103:
11098:
11093:
11092:
11090:
11085:
11080:
11077:
11071:
11064:
11060:
11057:
11053:
11052:
11047:
11042:
11037:
11034:
11030:
11029:
11027:
11002:
11001:
10990:
10985:
10977:
10974:
10969:
10964:
10960:
10957:
10952:
10949:
10942:
10939:
10934:
10929:
10926:
10925:
10921:
10917:
10913:
10910:
10905:
10900:
10899:
10897:
10892:
10887:
10884:
10878:
10871:
10867:
10864:
10860:
10859:
10855:
10851:
10848:
10844:
10843:
10841:
10789:
10788:
10779:
10777:
10766:
10761:
10754:
10750:
10746:
10743:
10738:
10732:
10728:
10725:
10724:
10719:
10716:
10711:
10705:
10701:
10698:
10695:
10691:
10690:
10688:
10681:
10673:
10670:
10665:
10660:
10654:
10651:
10646:
10640:
10636:
10632:
10627:
10622:
10619:
10615:
10614:
10610:
10606:
10602:
10599:
10594:
10589:
10583:
10580:
10575:
10569:
10565:
10561:
10556:
10551:
10550:
10548:
10543:
10538:
10535:
10529:
10522:
10518:
10515:
10511:
10510:
10506:
10502:
10499:
10495:
10494:
10492:
10442:
10441:
10432:
10430:
10413:
10410:
10405:
10400:
10394:
10391:
10386:
10383:
10378:
10374:
10369:
10364:
10361:
10359:
10355:
10352:
10347:
10344:
10337:
10334:
10329:
10324:
10318:
10315:
10310:
10307:
10302:
10298:
10293:
10287:
10281:
10278:
10273:
10268:
10263:
10260:
10255:
10250:
10249:
10244:
10241:
10236:
10233:
10226:
10223:
10218:
10213:
10207:
10204:
10199:
10196:
10191:
10187:
10182:
10177:
10174:
10172:
10168:
10165:
10160:
10155:
10149:
10146:
10141:
10138:
10133:
10129:
10124:
10118:
10112:
10109:
10104:
10099:
10098:
10093:
10090:
10085:
10080:
10074:
10071:
10066:
10063:
10058:
10054:
10049:
10043:
10037:
10034:
10029:
10024:
10021:
10019:
10015:
10012:
10007:
10004:
9997:
9994:
9989:
9984:
9978:
9975:
9970:
9967:
9962:
9958:
9953:
9948:
9947:
9942:
9939:
9934:
9931:
9924:
9921:
9916:
9911:
9905:
9902:
9897:
9894:
9889:
9885:
9880:
9874:
9868:
9865:
9860:
9855:
9850:
9847:
9842:
9837:
9834:
9832:
9828:
9825:
9820:
9815:
9809:
9806:
9801:
9798:
9793:
9789:
9784:
9779:
9778:
9748:
9747:
9738:
9736:
9725:
9720:
9712:
9709:
9704:
9701:
9694:
9691:
9686:
9681:
9675:
9672:
9667:
9664:
9659:
9655:
9650:
9644:
9638:
9635:
9630:
9625:
9620:
9617:
9612:
9606:
9602:
9599:
9594:
9589:
9583:
9580:
9575:
9572:
9567:
9563:
9558:
9552:
9546:
9543:
9538:
9533:
9530:
9529:
9524:
9521:
9516:
9513:
9506:
9503:
9498:
9493:
9487:
9484:
9479:
9476:
9471:
9467:
9462:
9457:
9454:
9450:
9447:
9442:
9437:
9431:
9428:
9423:
9420:
9415:
9411:
9406:
9401:
9400:
9398:
9393:
9388:
9385:
9379:
9372:
9368:
9365:
9361:
9360:
9356:
9352:
9349:
9345:
9344:
9342:
9250:matrices, and
9153:
9152:
9143:
9141:
9130:
9125:
9117:
9114:
9109:
9104:
9098:
9095:
9090:
9087:
9082:
9078:
9073:
9068:
9064:
9061:
9056:
9053:
9046:
9043:
9038:
9033:
9027:
9024:
9019:
9016:
9011:
9007:
9002:
8997:
8994:
8993:
8988:
8985:
8980:
8975:
8969:
8966:
8961:
8958:
8953:
8949:
8944:
8938:
8932:
8929:
8924:
8919:
8916:
8912:
8909:
8904:
8901:
8894:
8891:
8886:
8881:
8875:
8872:
8867:
8864:
8859:
8855:
8850:
8844:
8838:
8835:
8830:
8825:
8820:
8817:
8812:
8807:
8806:
8804:
8799:
8794:
8791:
8785:
8778:
8774:
8771:
8767:
8766:
8762:
8758:
8755:
8751:
8750:
8748:
8724:
8721:
8720:
8719:
8708:
8704:
8698:
8693:
8688:
8684:
8680:
8677:
8672:
8667:
8662:
8658:
8654:
8649:
8644:
8639:
8636:
8633:
8630:
8625:
8621:
8616:
8612:
8609:
8606:
8602:
8597:
8591:
8588:
8583:
8579:
8574:
8570:
8565:
8560:
8555:
8552:
8549:
8546:
8543:
8540:
8535:
8530:
8525:
8522:
8519:
8515:
8511:
8508:
8505:
8502:
8497:
8493:
8488:
8484:
8481:
8478:
8474:
8468:
8465:
8459:
8452:
8448:
8444:
8441:
8437:
8432:
8427:
8424:
8419:
8389:
8386:
8385:
8384:
8373:
8370:
8365:
8360:
8355:
8350:
8345:
8340:
8337:
8331:
8327:
8319:
8314:
8309:
8304:
8299:
8294:
8289:
8286:
8280:
8276:
8270:
8265:
8262:
8235:
8234:
8223:
8218:
8213:
8208:
8203:
8198:
8193:
8190:
8185:
8180:
8175:
8172:
8168:
8164:
8161:
8137:
8131:
8128:
8123:
8118:
8114:
8089:
8088:
8077:
8074:
8069:
8064:
8059:
8054:
8049:
8044:
8041:
8036:
8031:
8026:
8023:
8019:
8015:
8012:
7998:parallelepiped
7992:
7983:
7974:
7953:
7952:
7941:
7936:
7927:
7921:
7916:
7911:
7906:
7901:
7896:
7891:
7886:
7885:
7879:
7873:
7868:
7863:
7858:
7853:
7848:
7843:
7838:
7837:
7831:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7789:
7787:
7779:
7775:
7771:
7768:
7764:
7759:
7754:
7751:
7746:
7719:
7714:
7690:
7685:
7661:
7656:
7632:
7624:
7619:
7614:
7610:
7605:
7600:
7596:
7591:
7586:
7585:
7583:
7578:
7574:
7564:. If a matrix
7562:triple product
7550:
7549:
7538:
7534:
7528:
7523:
7518:
7514:
7510:
7507:
7503:
7499:
7495:
7490:
7486:
7481:
7476:
7471:
7468:
7465:
7462:
7457:
7453:
7448:
7444:
7441:
7438:
7434:
7428:
7425:
7419:
7412:
7408:
7404:
7401:
7397:
7392:
7387:
7384:
7379:
7363:
7362:
7351:
7348:
7345:
7342:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7317:
7313:
7310:
7296:rule of Sarrus
7286:
7285:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7242:
7239:
7237:
7235:
7232:
7229:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7196:
7193:
7191:
7189:
7186:
7183:
7181:
7178:
7175:
7172:
7169:
7166:
7163:
7160:
7157:
7153:
7150:
7148:
7146:
7143:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7115:
7112:
7108:
7105:
7103:
7101:
7098:
7095:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7065:
7062:
7060:
7058:
7055:
7052:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7019:
7016:
7014:
7012:
7009:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6984:
6981:
6977:
6974:
6972:
6970:
6967:
6964:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6931:
6928:
6926:
6924:
6921:
6918:
6916:
6913:
6910:
6907:
6904:
6901:
6898:
6895:
6892:
6888:
6885:
6883:
6881:
6878:
6877:
6847:
6846:
6833:
6827:
6823:
6821:
6817:
6815:
6811:
6810:
6807:
6803:
6801:
6797:
6795:
6791:
6790:
6787:
6783:
6781:
6777:
6775:
6771:
6770:
6768:
6760:
6756:
6752:
6749:
6745:
6740:
6734:
6727:
6721:
6717:
6715:
6711:
6709:
6705:
6704:
6701:
6697:
6695:
6691:
6689:
6685:
6684:
6681:
6677:
6675:
6671:
6669:
6665:
6664:
6662:
6653:
6649:
6645:
6642:
6638:
6633:
6628:
6625:
6619:
6613:
6610:
6608:
6605:
6603:
6600:
6599:
6596:
6593:
6591:
6588:
6586:
6583:
6582:
6579:
6576:
6574:
6571:
6569:
6566:
6565:
6563:
6557:
6552:
6549:
6544:
6520:
6517:
6516:
6515:
6504:
6500:
6495:
6491:
6487:
6482:
6477:
6473:
6470:
6466:
6461:
6453:
6449:
6445:
6440:
6435:
6432:
6427:
6392:
6391:
6380:
6375:
6369:
6365:
6363:
6360:
6357:
6356:
6353:
6350:
6345:
6343:
6337:
6336:
6334:
6326:
6323:
6320:
6317:
6314:
6310:
6305:
6300:
6294:
6290:
6288:
6285:
6282:
6281:
6278:
6275:
6270:
6268:
6262:
6261:
6259:
6250:
6246:
6242:
6237:
6232:
6229:
6223:
6217:
6214:
6212:
6209:
6208:
6205:
6202:
6200:
6197:
6196:
6194:
6188:
6183:
6180:
6175:
6150:
6147:
6103:
6102:
6090:
6085:
6082:
6077:
6072:
6064:
6057:
6053:
6052:
6050:
6045:
6040:
6035:
6032:
6027:
6021:
6015:
6010:
6001:
5994:
5990:
5989:
5987:
5982:
5977:
5972:
5969:
5964:
5959:
5956:
5951:
5946:
5930:
5929:
5916:
5908:
5905:
5900:
5895:
5893:
5890:
5886:
5883:
5878:
5873:
5869:
5866:
5861:
5856:
5855:
5852:
5849:
5847:
5844:
5842:
5839:
5837:
5834:
5833:
5828:
5825:
5820:
5815:
5813:
5810:
5806:
5801:
5796:
5792:
5787:
5782:
5781:
5776:
5773:
5768:
5763:
5761:
5758:
5754:
5749:
5744:
5740:
5735:
5730:
5729:
5727:
5718:
5711:
5707:
5706:
5704:
5699:
5694:
5688:
5682:
5673:
5666:
5662:
5661:
5659:
5654:
5649:
5644:
5641:
5636:
5610:, known as an
5600:Main article:
5597:
5594:
5554:
5553:
5542:
5537:
5534:
5529:
5522:
5519:
5514:
5509:
5504:
5499:
5494:
5489:
5486:
5481:
5451:Main article:
5448:
5445:
5444:
5443:
5432:
5425:
5421:
5417:
5412:
5407:
5404:
5399:
5394:
5391:
5387:
5383:
5352:
5346:
5340:
5335:
5330:
5327:
5322:
5301:is symmetric,
5284:
5279:
5275:
5271:
5266:
5263:
5259:
5220:
5216:
5187:is the square
5179:
5178:
5167:
5162:
5159:
5154:
5147:
5144:
5139:
5133:
5129:
5124:
5121:
5116:
5084:Main article:
5081:
5078:
5070:
5069:
5058:
5055:
5050:
5047:
5044:
5040:
5036:
5033:
5030:
5025:
5021:
5017:
5012:
5008:
5004:
4999:
4996:
4993:
4989:
4982:
4979:
4976:
4973:
4970:
4967:
4960:
4957:
4954:
4950:
4946:
4943:
4940:
4932:
4929:
4926:
4921:
4914:
4909:
4906:
4903:
4899:
4892:
4888:
4884:
4881:
4877:
4872:
4867:
4864:
4859:
4833:
4828:
4824:
4820:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4784:
4780:
4764:
4763:
4752:
4749:
4746:
4743:
4738:
4734:
4730:
4725:
4722:
4719:
4714:
4711:
4708:
4704:
4700:
4697:
4671:
4668:
4663:
4659:
4611:
4610:
4599:
4592:
4588:
4582:
4577:
4572:
4567:
4562:
4559:
4553:
4548:
4544:
4536:
4532:
4527:
4519:
4516:
4511:
4507:
4502:
4498:
4495:
4492:
4484:
4481:
4478:
4473:
4470:
4467:
4463:
4455:
4452:
4449:
4445:
4441:
4438:
4435:
4430:
4426:
4422:
4417:
4413:
4408:
4402:
4397:
4390:
4387:
4384:
4379:
4376:
4373:
4369:
4362:
4358:
4354:
4351:
4347:
4342:
4337:
4334:
4329:
4288:
4285:
4254:
4253:
4242:
4237:
4233:
4229:
4224:
4220:
4216:
4211:
4207:
4203:
4200:
4195:
4192:
4189:
4185:
4168:as used for a
4161:
4158:
4117:
4113:
4107:
4102:
4095:
4090:
4085:
4080:
4077:
4074:
4069:
4062:
4057:
4032:
4027:
4024:
4019:
4014:
4009:
4006:
4001:
3995:
3974:
3969:
3966:
3961:
3955:
3951:
3947:
3941:
3936:
3929:
3924:
3919:
3914:
3911:
3908:
3903:
3896:
3891:
3869:
3864:
3861:
3856:
3832:
3828:
3824:
3820:
3814:
3809:
3802:
3797:
3792:
3787:
3784:
3781:
3776:
3769:
3764:
3740:
3735:
3707:
3702:
3696:
3693:
3691:
3688:
3687:
3684:
3681:
3679:
3676:
3675:
3673:
3668:
3663:
3660:
3655:
3631:
3627:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3601:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3579:
3576:
3555:
3552:
3547:
3543:
3539:
3534:
3530:
3525:
3522:
3502:
3497:
3493:
3489:
3484:
3480:
3476:
3473:
3451:
3447:
3440:
3437:
3435:
3432:
3427:
3424:
3418:
3416:
3413:
3412:
3409:
3406:
3403:
3401:
3398:
3395:
3393:
3390:
3388:
3385:
3382:
3381:
3378:
3357:
3354:
3349:
3345:
3341:
3336:
3332:
3327:
3324:
3319:
3315:
3311:
3289:
3285:
3278:
3275:
3273:
3270:
3265:
3262:
3256:
3254:
3251:
3250:
3247:
3244:
3242:
3239:
3234:
3231:
3225:
3223:
3220:
3217:
3216:
3213:
3192:
3189:
3184:
3180:
3176:
3171:
3167:
3163:
3158:
3154:
3150:
3128:
3124:
3101:
3097:
3074:
3070:
3063:
3060:
3058:
3055:
3053:
3050:
3047:
3045:
3042:
3041:
3038:
3035:
3033:
3030:
3025:
3022:
3016:
3014:
3011:
3008:
3007:
3004:
2981:
2976:
2970:
2967:
2964:
2962:
2959:
2958:
2952:
2949:
2943:
2941:
2938:
2935:
2934:
2932:
2927:
2923:
2896:
2893:
2891:
2888:
2870:
2858:
2857:
2846:
2841:
2835:
2832:
2829:
2824:
2821:
2815:
2814:
2808:
2805:
2799:
2797:
2794:
2791:
2790:
2788:
2783:
2779:
2750:
2747:
2742:
2739:
2735:
2731:
2710:
2698:
2697:
2686:
2681:
2675:
2672:
2669:
2667:
2664:
2663:
2657:
2654:
2648:
2646:
2643:
2640:
2639:
2637:
2632:
2628:
2600:
2599:
2588:
2583:
2577:
2574:
2572:
2569:
2568:
2565:
2562:
2560:
2557:
2556:
2554:
2549:
2545:
2522:
2519:
2447:measure theory
2409:
2404:
2401:
2398:
2393:
2365:
2362:
2361:
2360:
2345:
2341:
2337:
2334:
2303:
2302:
2287:
2283:
2279:
2276:
2256:
2253:
2252:
2251:
2240:
2237:
2233:
2229:
2226:
2223:
2217:
2213:
2209:
2206:
2202:
2197:
2192:
2189:
2184:
2142:
2139:
2073:
2068:
2065:
2062:
2058:
2054:
2049:
2045:
2038:
2032:
2028:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1960:
1956:
1926:
1920:
1916:
1862:
1861:
1850:
1845:
1842:
1838:
1833:
1829:
1826:
1823:
1818:
1815:
1810:
1805:
1795:
1784:
1779:
1776:
1771:
1766:
1759:
1756:
1751:
1746:
1741:
1736:
1733:
1728:
1725:
1722:
1717:
1710:
1707:
1702:
1697:
1692:
1687:
1684:
1680:
1674:
1669:
1662:
1659:
1656:
1651:
1646:
1641:
1636:
1629:
1624:
1619:
1589:
1584:
1579:
1576:
1573:
1568:
1563:
1541:
1536:
1533:
1528:
1521:
1518:
1513:
1508:
1503:
1500:
1496:
1491:
1488:
1484:
1453:
1439:
1434:
1428:
1425:
1420:
1415:
1412:
1407:
1404:
1400:
1393:
1387:
1382:
1372:
1342:
1339:
1334:
1327:
1322:
1317:
1312:
1308:
1303:
1300:
1296:
1286:
1269:
1266:
1261:
1254:
1251:
1247:
1243:
1238:
1235:
1231:
1226:
1222:
1219:
1209:
1197:
1193:
1188:
1185:
1181:
1175:
1172:
1167:
1162:
1141:
1138:
1137:
1136:
1123:
1104:
1100:
1097:
1093:
1083:if the matrix
1066:
1045:
1016:
1003:
986:
943:
927:has a trivial
920:
897:
882:
875:
849:
842:
825:row-equivalent
816:
804:
789:
778:
740:
721:matrix over a
704:
701:
699:
696:
683:
633:multiplication
627:equipped with
606:
555:
454:"almost never"
434:if and only if
355:
339:
334:
329:
324:
320:
317:
313:
309:
306:
290:square matrix
248:linear algebra
242:
241:
224:
223:
185:
183:
176:
169:
168:
151:September 2020
83:
81:
74:
69:
43:
42:
40:
33:
26:
9:
6:
4:
3:
2:
18311:
18300:
18299:Matrix theory
18297:
18295:
18292:
18290:
18287:
18285:
18282:
18281:
18279:
18264:
18261:
18259:
18256:
18254:
18253:
18248:
18242:
18241:
18238:
18232:
18229:
18227:
18224:
18222:
18221:Pseudoinverse
18219:
18217:
18214:
18212:
18209:
18207:
18204:
18202:
18199:
18197:
18194:
18193:
18191:
18189:Related terms
18187:
18181:
18180:Z (chemistry)
18178:
18176:
18173:
18171:
18168:
18166:
18163:
18161:
18158:
18156:
18153:
18151:
18148:
18146:
18143:
18141:
18138:
18136:
18133:
18131:
18128:
18126:
18123:
18121:
18118:
18117:
18115:
18111:
18105:
18102:
18100:
18097:
18095:
18092:
18090:
18087:
18085:
18082:
18080:
18077:
18075:
18072:
18070:
18067:
18066:
18064:
18062:
18057:
18051:
18048:
18046:
18043:
18041:
18038:
18036:
18033:
18031:
18028:
18026:
18023:
18021:
18018:
18016:
18013:
18011:
18008:
18006:
18003:
18002:
18000:
17998:
17993:
17987:
17984:
17982:
17979:
17977:
17974:
17972:
17969:
17967:
17964:
17962:
17959:
17957:
17954:
17952:
17949:
17947:
17944:
17942:
17939:
17937:
17934:
17932:
17929:
17927:
17924:
17922:
17919:
17917:
17914:
17912:
17909:
17907:
17904:
17902:
17899:
17897:
17894:
17892:
17889:
17887:
17884:
17882:
17879:
17877:
17874:
17872:
17869:
17867:
17864:
17862:
17859:
17857:
17854:
17852:
17849:
17847:
17844:
17842:
17839:
17837:
17834:
17833:
17831:
17827:
17821:
17818:
17816:
17813:
17811:
17808:
17806:
17803:
17801:
17798:
17796:
17793:
17791:
17788:
17786:
17783:
17781:
17778:
17776:
17773:
17771:
17767:
17764:
17762:
17759:
17758:
17756:
17754:
17750:
17745:
17739:
17736:
17734:
17731:
17729:
17726:
17724:
17721:
17719:
17716:
17714:
17711:
17709:
17706:
17704:
17701:
17700:
17698:
17696:
17691:
17685:
17682:
17680:
17677:
17675:
17672:
17670:
17667:
17665:
17662:
17660:
17657:
17655:
17652:
17650:
17647:
17645:
17642:
17640:
17637:
17636:
17634:
17630:
17624:
17621:
17619:
17616:
17614:
17611:
17609:
17606:
17604:
17601:
17599:
17596:
17594:
17591:
17589:
17586:
17584:
17581:
17579:
17576:
17574:
17571:
17569:
17566:
17564:
17561:
17559:
17556:
17554:
17551:
17549:
17546:
17544:
17541:
17539:
17538:Pentadiagonal
17536:
17534:
17531:
17529:
17526:
17524:
17521:
17519:
17516:
17514:
17511:
17509:
17506:
17504:
17501:
17499:
17496:
17494:
17491:
17489:
17486:
17484:
17481:
17479:
17476:
17474:
17471:
17469:
17466:
17464:
17461:
17459:
17456:
17454:
17451:
17449:
17446:
17444:
17441:
17439:
17436:
17434:
17431:
17429:
17426:
17424:
17421:
17419:
17416:
17414:
17411:
17409:
17406:
17404:
17401:
17399:
17396:
17394:
17391:
17389:
17386:
17384:
17381:
17379:
17376:
17374:
17371:
17369:
17368:Anti-diagonal
17366:
17364:
17361:
17360:
17358:
17354:
17349:
17342:
17337:
17335:
17330:
17328:
17323:
17322:
17319:
17307:
17299:
17298:
17295:
17289:
17286:
17284:
17283:Sparse matrix
17281:
17279:
17276:
17274:
17271:
17269:
17266:
17265:
17263:
17261:
17257:
17251:
17248:
17246:
17243:
17241:
17238:
17236:
17233:
17231:
17228:
17226:
17223:
17222:
17220:
17218:constructions
17217:
17213:
17207:
17206:Outermorphism
17204:
17202:
17199:
17197:
17194:
17192:
17189:
17187:
17184:
17182:
17179:
17177:
17174:
17172:
17169:
17167:
17166:Cross product
17164:
17162:
17159:
17158:
17156:
17154:
17150:
17144:
17141:
17139:
17136:
17134:
17133:Outer product
17131:
17129:
17126:
17124:
17121:
17119:
17116:
17114:
17113:Orthogonality
17111:
17110:
17108:
17106:
17102:
17096:
17093:
17091:
17090:Cramer's rule
17088:
17086:
17083:
17081:
17078:
17076:
17073:
17071:
17068:
17066:
17063:
17061:
17060:Decomposition
17058:
17056:
17053:
17052:
17050:
17048:
17044:
17039:
17029:
17026:
17024:
17021:
17019:
17016:
17014:
17011:
17009:
17006:
17004:
17001:
16999:
16996:
16994:
16991:
16989:
16986:
16984:
16981:
16979:
16976:
16974:
16971:
16969:
16966:
16964:
16961:
16959:
16956:
16954:
16951:
16949:
16946:
16944:
16941:
16939:
16936:
16935:
16933:
16929:
16923:
16920:
16918:
16915:
16914:
16911:
16907:
16900:
16895:
16893:
16888:
16886:
16881:
16880:
16877:
16871:
16868:
16864:
16863:
16858:
16853:
16849:
16845:
16841:
16837:
16832:
16831:
16824:
16821:
16813:
16803:
16799:
16798:inappropriate
16795:
16791:
16785:
16783:
16776:
16767:
16766:
16751:
16746:
16742:
16738:
16732:
16728:
16727:
16721:
16717:
16715:0-262-03293-7
16711:
16707:
16706:
16701:
16697:
16693:
16689:
16685:
16681:
16677:
16676:
16671:
16667:
16666:
16652:
16647:
16643:
16639:
16635:
16628:
16620:
16614:
16610:
16606:
16602:
16596:
16588:
16586:0-471-98633-X
16582:
16578:
16571:
16556:
16552:
16546:
16537:
16532:
16528:
16524:
16520:
16516:
16512:
16505:
16497:
16491:
16487:
16480:
16473:
16469:
16465:
16460:
16453:
16447:
16445:
16436:
16430:
16426:
16419:
16411:
16405:
16401:
16394:
16386:
16382:
16378:
16374:
16367:
16361:
16355:
16349:
16345:
16344:
16336:
16328:
16324:
16320:
16316:
16312:
16308:
16305:(1): 99–115.
16304:
16300:
16296:
16288:
16281:
16277:
16270:
16263:
16259:
16252:
16243:
16237:
16233:
16229:
16222:
16208:
16204:
16197:
16189:
16182:
16174:
16168:
16164:
16160:
16156:
16152:
16146:
16142:
16131:
16128:
16126:
16123:
16121:
16118:
16116:
16115:Pseudoinverse
16113:
16111:
16108:
16106:
16103:
16101:
16098:
16096:
16093:
16091:
16088:
16086:
16083:
16082:
16075:
16073:
16069:
16065:
16062: ×
16061:
16057:
16053:
16049:
16045:
16041:
16037:
16034:transmit and
16033:
16029:
16025:
16015:
16013:
16009:
16005:
15995:
15986:
15984:
15979:
15977:
15973:
15969:
15959:
15957:
15953:
15949:
15945:
15917:
15913:
15908:
15904:
15900:
15889:
15881:
15878:
15875:
15872:
15869:
15863:
15846:
15843:
15831:
15826:
15823:
15820:
15816:
15812:
15809:
15804:
15801:
15791:
15789:
15782:
15779:
15766:
15763:
15748:
15744:
15739:
15735:
15731:
15720:
15715:
15712:
15709:
15692:
15689:
15686:
15674:
15669:
15666:
15663:
15659:
15655:
15652:
15647:
15637:
15635:
15628:
15615:
15612:
15593:
15592:
15591:
15570:
15562:
15559:
15556:
15553:
15550:
15544:
15531:
15506:
15503:
15491:
15486:
15483:
15480:
15476:
15472:
15469:
15467:
15459:
15447:
15444:
15422:
15417:
15414:
15411:
15398:
15373:
15370:
15367:
15355:
15350:
15347:
15344:
15340:
15336:
15334:
15326:
15314:
15288:
15287:
15286:
15272:
15249:
15245:
15240:
15236:
15232:
15221:
15205:
15201:
15179:
15175:
15169:
15165:
15161:
15158:
15144:
15141:
15130:
15127:
15116:
15109:
15108:
15107:
15090:
15074:
15070:
15063:
15027:
15023:
15017:
15013:
15009:
15003:
14982:
14967:
14966:
14965:
14948:
14939:
14935:
14921:
14916:
14913:
14896:
14893:
14883:
14880:
14875:
14872:
14862:
14857:
14854:
14849:
14840:
14837:
14828:
14819:
14818:
14817:
14803:
14780:
14775:
14772:
14759:
14734:
14731:
14721:
14718:
14712:
14700:
14697:
14675:
14674:
14673:
14657:
14654:
14624:
14599:
14596:
14566:
14558:
14552:
14529:
14523:
14498:
14495:
14485:
14474:
14462:
14459:
14441:
14435:
14415:
14412:
14387:
14386:
14385:
14383:
14362:
14352:
14349:
14334:
14315:
14310:
14307:
14294:
14269:
14266:
14256:
14253:
14247:
14235:
14232:
14210:
14209:
14208:
14206:
14202:
14198:
14194:
14184:
14148:
14145:
14137:
14127:
14124:
14121:
14116:
14106:
14101:
14069:
14040:
14030:
14026:
14021:
14016:
14012:
14008:
14004:
14000:
13994:
13984:
13979:
13968:
13964:
13960:
13955:
13949:
13939:
13934:
13923:
13918:
13912:
13902:
13897:
13886:
13881:
13877:
13867:
13864:
13849:
13831:
13826:
13822:
13799:
13789:
13785:
13780:
13775:
13771:
13767:
13763:
13759:
13753:
13743:
13738:
13727:
13723:
13718:
13715:
13681:
13652:
13623:
13595:
13587:
13584:
13580:
13573:
13568:
13565:
13533:
13530:
13520:
13510:
13507:
13504:
13499:
13486:
13481:
13468:
13460:
13450:
13447:
13444:
13439:
13428:
13425:
13422:
13417:
13402:
13399:
13396:
13388:
13385:
13379:
13374:
13362:
13359:
13355:
13351:
13346:
13332:
13331:
13330:
13328:
13324:
13320:
13302:
13297:
13293:
13289:
13284:
13274:
13269:
13259:
13254:
13244:
13239:
13210:
13196:
13192:
13174:
13159:
13141:
13129:
13126:
13122:
13118:
13113:
13078:
13075:
13072:
13069:
13066:
13063:
13060:
13039:
13034:
13031:
13027:
13023:
13019:
13005:
13001:
12994:
12984:
12980:
12976:
12960:
12952:
12948:
12942:
12935:
12931:
12927:
12922:
12913:
12894:
12884:
12872:
12859:
12856:
12845:
12841:
12838:
12835:
12832:
12825:
12822:
12804:
12791:
12788:
12775:
12770:
12767:
12757:
12752:
12749:
12735:
12734:
12733:
12731:
12726:
12722:
12702:
12694:
12691:
12679:
12674:
12662:
12649:
12646:
12635:
12623:
12620:
12617:
12613:
12609:
12604:
12601:
12587:
12586:
12585:
12582:
12562:
12559:
12554:
12549:
12543:
12540:
12525:
12516:
12503:
12475:
12472:
12467:
12462:
12451:
12448:
12438:
12429:
12416:
12404:
12403:
12402:
12399:
12393:
12387:
12379:
12356:
12348:
12344:
12330:
12319:
12310:
12304:
12301:
12298:
12293:
12290:
12287:
12283:
12279:
12274:
12261:
12248:
12245:
12240:
12236:
12230:
12227:
12224:
12220:
12212:
12211:
12210:
12208:
12207:geometric sum
12204:
12185:
12180:
12167:
12149:
12146:
12143:
12139:
12135:
12130:
12127:
12113:
12112:
12111:
12109:
12104:
12084:
12081:
12076:
12063:
12044:
12032:
12031:
12030:
12027:
12016:
12012:
12008:
12004:
11997:
11993:
11988:
11984:
11980:
11975:
11972:
11948:
11934:
11916:
11904:
11901:
11896:
11885:
11874:
11869:
11864:
11861:
11805:
11756:
11751:
11748:
11731:
11728:
11716:
11704:
11701:
11691:
11686:
11674:
11669:
11659:
11654:
11628:
11623:
11620:
11603:
11600:
11590:
11585:
11573:
11570:
11560:
11555:
11529:
11524:
11512:
11509:
11494:
11489:
11477:
11467:
11462:
11436:
11431:
11428:
11413:
11408:
11392:
11386:
11367:
11355:
11352:
11337:
11329:
11311:
11304:
11302:
11288:
11283:
11275:
11272:
11258:
11255:
11238:
11235:
11225:
11216:
11213:
11201:
11189:
11186:
11176:
11169:
11166:
11149:
11146:
11134:
11122:
11119:
11109:
11104:
11101:
11088:
11083:
11078:
11075:
11069:
11045:
11025:
11015:
11014:
11011:
11008:
10988:
10983:
10975:
10972:
10958:
10955:
10940:
10937:
10927:
10911:
10908:
10895:
10890:
10885:
10882:
10876:
10839:
10829:
10828:
10827:
10825:
10820:
10814:
10809:
10804:
10798:
10796:
10787:
10780:
10778:
10764:
10759:
10744:
10741:
10726:
10717:
10714:
10699:
10686:
10679:
10671:
10668:
10663:
10652:
10649:
10634:
10625:
10600:
10597:
10592:
10581:
10578:
10563:
10554:
10546:
10541:
10536:
10533:
10527:
10490:
10480:
10479:
10476:
10473:
10467:
10461:
10459:
10455:
10451:
10450:
10440:
10433:
10431:
10411:
10408:
10403:
10392:
10389:
10376:
10367:
10362:
10360:
10353:
10350:
10335:
10332:
10327:
10316:
10313:
10300:
10291:
10279:
10276:
10266:
10261:
10258:
10242:
10239:
10224:
10221:
10216:
10205:
10202:
10189:
10180:
10175:
10173:
10166:
10163:
10158:
10147:
10144:
10131:
10122:
10110:
10107:
10091:
10088:
10083:
10072:
10069:
10056:
10047:
10035:
10032:
10022:
10020:
10013:
10010:
9995:
9992:
9987:
9976:
9973:
9960:
9951:
9940:
9937:
9922:
9919:
9914:
9903:
9900:
9887:
9878:
9866:
9863:
9853:
9848:
9845:
9835:
9833:
9826:
9823:
9818:
9807:
9804:
9791:
9782:
9769:
9768:
9765:
9763:
9762:
9757:
9756:
9746:
9739:
9737:
9723:
9718:
9710:
9707:
9692:
9689:
9684:
9673:
9670:
9657:
9648:
9636:
9633:
9623:
9618:
9615:
9600:
9597:
9592:
9581:
9578:
9565:
9556:
9544:
9541:
9531:
9522:
9519:
9504:
9501:
9496:
9485:
9482:
9469:
9460:
9455:
9448:
9445:
9440:
9429:
9426:
9413:
9404:
9396:
9391:
9386:
9383:
9377:
9340:
9330:
9329:
9326:
9323:
9320:
9316:
9310:
9304:
9298:
9292:
9286:
9281:
9280:
9274:
9271:
9265:
9260:
9255:
9253:
9249:
9245:
9240:
9237:
9232:
9227:
9224:
9220:
9214:
9208:
9205:
9201:
9195:
9189:
9184:
9179:
9173:
9167:
9161:
9151:
9144:
9142:
9128:
9123:
9115:
9112:
9107:
9096:
9093:
9080:
9071:
9062:
9059:
9044:
9041:
9036:
9025:
9022:
9009:
9000:
8995:
8986:
8983:
8978:
8967:
8964:
8951:
8942:
8930:
8927:
8917:
8910:
8907:
8892:
8889:
8884:
8873:
8870:
8857:
8848:
8836:
8833:
8823:
8818:
8815:
8802:
8797:
8792:
8789:
8783:
8746:
8736:
8735:
8732:
8730:
8706:
8702:
8696:
8686:
8678:
8675:
8670:
8660:
8656:
8647:
8634:
8631:
8628:
8623:
8610:
8607:
8600:
8589:
8586:
8581:
8572:
8563:
8550:
8547:
8544:
8541:
8533:
8520:
8517:
8509:
8506:
8503:
8500:
8495:
8482:
8479:
8472:
8466:
8463:
8457:
8435:
8430:
8425:
8422:
8408:
8407:
8406:
8402:
8396:
8371:
8363:
8353:
8348:
8335:
8312:
8302:
8297:
8284:
8268:
8263:
8260:
8253:
8252:
8251:
8248:
8245:
8241:
8216:
8206:
8201:
8188:
8183:
8173:
8152:
8151:
8150:
8129:
8126:
8116:
8102:
8096:
8075:
8067:
8057:
8052:
8039:
8034:
8024:
8003:
8002:
8001:
7999:
7991:
7982:
7973:
7966:
7959:
7939:
7934:
7914:
7904:
7899:
7866:
7856:
7851:
7818:
7808:
7803:
7785:
7762:
7757:
7752:
7749:
7735:
7734:
7733:
7717:
7688:
7659:
7630:
7622:
7608:
7594:
7581:
7576:
7563:
7559:
7558:cross product
7536:
7532:
7526:
7516:
7508:
7505:
7497:
7488:
7479:
7466:
7463:
7460:
7455:
7442:
7439:
7432:
7426:
7423:
7417:
7395:
7390:
7385:
7382:
7368:
7367:
7366:
7349:
7346:
7343:
7340:
7337:
7334:
7331:
7328:
7325:
7322:
7301:
7300:
7299:
7297:
7292:
7268:
7262:
7259:
7256:
7253:
7250:
7240:
7238:
7233:
7225:
7219:
7216:
7213:
7210:
7207:
7201:
7194:
7192:
7187:
7179:
7173:
7170:
7167:
7164:
7161:
7151:
7149:
7144:
7137:
7131:
7128:
7125:
7122:
7119:
7113:
7106:
7104:
7099:
7091:
7085:
7082:
7079:
7076:
7073:
7063:
7061:
7056:
7048:
7042:
7039:
7036:
7033:
7030:
7024:
7017:
7015:
7010:
7003:
6997:
6994:
6991:
6988:
6985:
6975:
6973:
6968:
6960:
6954:
6951:
6948:
6945:
6942:
6936:
6929:
6927:
6922:
6914:
6908:
6905:
6902:
6899:
6896:
6886:
6884:
6879:
6868:
6867:
6866:
6863:
6860:
6852:
6831:
6825:
6819:
6813:
6805:
6799:
6793:
6785:
6779:
6773:
6766:
6743:
6738:
6725:
6719:
6713:
6707:
6699:
6693:
6687:
6679:
6673:
6667:
6660:
6636:
6631:
6626:
6623:
6617:
6611:
6606:
6601:
6594:
6589:
6584:
6577:
6572:
6567:
6561:
6555:
6550:
6547:
6533:
6532:
6531:
6526:
6502:
6498:
6489:
6480:
6471:
6468:
6464:
6459:
6443:
6438:
6433:
6430:
6416:
6415:
6414:
6411:
6409:
6403:
6399:
6378:
6373:
6367:
6361:
6358:
6351:
6348:
6341:
6332:
6324:
6321:
6318:
6315:
6312:
6308:
6303:
6298:
6292:
6286:
6283:
6276:
6273:
6266:
6257:
6240:
6235:
6230:
6227:
6221:
6215:
6210:
6203:
6198:
6192:
6186:
6181:
6178:
6164:
6163:
6162:
6156:
6146:
6144:
6139:
6134:
6129:
6123:
6118:
6111:
6088:
6083:
6080:
6070:
6062:
6048:
6043:
6038:
6033:
6030:
6025:
6008:
5999:
5985:
5980:
5975:
5970:
5967:
5962:
5957:
5954:
5944:
5935:
5934:
5933:
5914:
5906:
5903:
5891:
5884:
5881:
5867:
5864:
5850:
5845:
5840:
5835:
5826:
5823:
5811:
5804:
5790:
5774:
5771:
5759:
5752:
5738:
5725:
5716:
5702:
5697:
5692:
5671:
5657:
5652:
5647:
5642:
5639:
5625:
5624:
5623:
5621:
5617:
5613:
5609:
5603:
5602:Cramer's rule
5593:
5590:
5585:
5579:
5573:
5568:
5565:
5560:
5540:
5535:
5532:
5520:
5517:
5512:
5507:
5497:
5492:
5487:
5484:
5470:
5469:
5468:
5466:
5461:
5454:
5430:
5423:
5419:
5415:
5410:
5405:
5402:
5397:
5392:
5389:
5381:
5372:
5371:
5370:
5367:
5350:
5333:
5328:
5325:
5310:
5305:
5299:
5282:
5277:
5273:
5269:
5264:
5261:
5248:
5243:
5237:
5218:
5214:
5206:
5199:matrix whose
5196:
5192:
5185:
5165:
5160:
5157:
5145:
5142:
5127:
5122:
5119:
5105:
5104:
5103:
5100:
5094:
5087:
5077:
5075:
5056:
5048:
5045:
5042:
5038:
5034:
5031:
5028:
5023:
5019:
5015:
5010:
5006:
4997:
4994:
4991:
4987:
4980:
4974:
4971:
4968:
4958:
4955:
4952:
4944:
4941:
4930:
4927:
4924:
4912:
4907:
4904:
4901:
4897:
4875:
4870:
4865:
4862:
4848:
4847:
4846:
4831:
4826:
4822:
4818:
4814:
4811:
4808:
4802:
4799:
4796:
4790:
4787:
4782:
4778:
4770:of arguments
4769:
4750:
4747:
4744:
4741:
4736:
4732:
4728:
4723:
4720:
4717:
4712:
4709:
4706:
4702:
4698:
4695:
4688:
4687:
4686:
4685:
4669:
4666:
4661:
4657:
4644:
4643:main diagonal
4639:
4634:
4628:
4621:
4597:
4590:
4586:
4580:
4575:
4565:
4560:
4557:
4551:
4546:
4542:
4534:
4530:
4525:
4517:
4514:
4509:
4505:
4496:
4493:
4482:
4479:
4476:
4471:
4468:
4465:
4461:
4453:
4450:
4447:
4443:
4439:
4436:
4433:
4428:
4424:
4420:
4415:
4411:
4406:
4400:
4388:
4385:
4382:
4377:
4374:
4371:
4367:
4345:
4340:
4335:
4332:
4318:
4317:
4316:
4313:
4306:
4299:
4294:
4284:
4282:
4278:
4273:
4269:
4264:
4262:
4258:
4240:
4235:
4231:
4227:
4222:
4218:
4214:
4209:
4205:
4201:
4198:
4193:
4190:
4187:
4183:
4175:
4174:
4173:
4171:
4167:
4157:
4154:
4148:
4143:
4140:and applying
4138:
4132:
4115:
4105:
4093:
4083:
4078:
4075:
4072:
4060:
4044:
4030:
4025:
4022:
4012:
4007:
4004:
3972:
3967:
3964:
3949:
3939:
3927:
3917:
3912:
3909:
3906:
3894:
3867:
3862:
3859:
3843:
3830:
3822:
3812:
3800:
3790:
3785:
3782:
3779:
3767:
3738:
3723:
3718:
3705:
3700:
3694:
3689:
3682:
3677:
3671:
3666:
3661:
3658:
3642:
3629:
3625:
3618:
3613:
3608:
3603:
3596:
3591:
3586:
3581:
3574:
3553:
3545:
3541:
3532:
3528:
3523:
3495:
3491:
3482:
3478:
3474:
3462:
3449:
3445:
3438:
3433:
3425:
3422:
3414:
3407:
3404:
3399:
3396:
3391:
3386:
3383:
3376:
3368:which yields
3355:
3347:
3343:
3334:
3330:
3325:
3322:
3317:
3313:
3300:
3287:
3283:
3276:
3271:
3263:
3260:
3252:
3245:
3240:
3232:
3229:
3221:
3218:
3211:
3190:
3182:
3178:
3169:
3165:
3161:
3156:
3152:
3126:
3122:
3099:
3095:
3085:
3072:
3068:
3061:
3056:
3051:
3048:
3043:
3036:
3031:
3023:
3020:
3012:
3009:
3002:
2992:
2979:
2974:
2968:
2965:
2960:
2950:
2947:
2939:
2936:
2930:
2925:
2911:
2909:
2905:
2901:
2887:
2885:
2844:
2839:
2833:
2830:
2822:
2819:
2806:
2803:
2795:
2792:
2786:
2781:
2769:
2768:
2767:
2764:
2748:
2745:
2740:
2737:
2684:
2679:
2673:
2670:
2665:
2655:
2652:
2644:
2641:
2635:
2630:
2618:
2617:
2616:
2613:
2610:
2606:
2586:
2581:
2575:
2570:
2563:
2558:
2552:
2547:
2535:
2534:
2533:
2529:
2518:
2516:
2512:
2507:
2497:
2496:nowhere dense
2493:
2481:
2477:
2474:
2461:
2452:
2448:
2444:
2440:
2436:
2432:
2429:
2425:
2407:
2402:
2399:
2396:
2379:
2339:
2324:
2323:
2322:
2319:
2313:
2308:
2307:finite square
2281:
2266:
2265:
2264:
2262:
2261:associativity
2238:
2224:
2221:
2200:
2195:
2190:
2187:
2173:
2172:
2171:
2168:
2162:
2159:
2153:
2148:
2138:
2136:
2131:
2127:
2121:
2117:
2111:
2105:
2102:
2097:
2092:
2087:
2071:
2066:
2063:
2060:
2056:
2052:
2047:
2043:
2030:
2026:
2017:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1958:
1954:
1944:
1918:
1914:
1904:
1898:
1894:
1890:
1884:
1879:
1874:
1868:
1848:
1843:
1840:
1821:
1816:
1813:
1796:
1782:
1777:
1774:
1769:
1757:
1754:
1749:
1739:
1734:
1731:
1726:
1723:
1720:
1708:
1705:
1700:
1690:
1685:
1682:
1672:
1660:
1657:
1654:
1644:
1639:
1627:
1587:
1577:
1574:
1571:
1566:
1539:
1534:
1531:
1519:
1516:
1506:
1501:
1498:
1473:
1467:
1454:
1426:
1423:
1410:
1405:
1402:
1373:
1369:
1364:
1359:
1340:
1337:
1325:
1315:
1310:
1287:
1267:
1264:
1252:
1249:
1245:
1241:
1236:
1233:
1220:
1210:
1191:
1186:
1183:
1173:
1170:
1153:
1152:
1151:
1148:
1134:
1129:
1124:
1120:
1098:
1095:
1081:
1064:
1055:
1050:
1046:
1043:
1039:
1033:
1026:
1021:
1017:
1009:
1004:
997:
992:
987:
984:
980:
971:
965:
961:
955:
949:
944:
940:
936:
930:
925:
921:
917:
913:
907:
902:
898:
895:
887:
883:
878:
874:
870:
859:
854:
850:
845:
841:
837:
826:
821:
817:
813:
809:
805:
801:
795:
790:
786:
781:
777:
773:
767:
761:
757:
756:
755:
724:
711:
695:
691:
686:
675:
671:
663:
658:
654:
648:
646:
642:
638:
634:
630:
626:
622:
618:
613:
609:
605:
601:
595:
581:
577:
571:
565:
558:
554:
550:
544:
538:
532:
526:
520:
516:
510:
504:
499:
486:
481:
480:right inverse
477:
472:
468:
455:
451:
447:
446:complex plane
443:
439:
435:
431:
427:
423:
419:
414:
412:
407:
403:, denoted by
401:
396:
395:
394:
386:
380:
375:
371:
358:
354:
337:
332:
322:
311:
294:
281:
277:
276:nondegenerate
273:
269:
264:
260:
259:square matrix
249:
238:
235:
220:
217:
209:
199:
195:
189:
186:This article
184:
175:
174:
165:
162:
154:
143:
140:
136:
133:
129:
126:
122:
119:
115:
112: –
111:
107:
106:Find sources:
100:
96:
90:
89:
84:This article
82:
78:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
18294:Determinants
18243:
18175:Substitution
18061:graph theory
17774:
17558:Quaternionic
17548:Persymmetric
17216:Vector space
17064:
16948:Vector space
16860:
16839:
16816:
16807:
16792:by removing
16779:
16741:Google Books
16739:– via
16725:
16704:
16673:
16641:
16637:
16627:
16604:
16595:
16576:
16570:
16558:. Retrieved
16554:
16545:
16518:
16514:
16504:
16485:
16479:
16459:
16451:
16424:
16418:
16399:
16393:
16376:
16372:
16366:
16342:
16335:
16302:
16298:
16287:
16275:
16269:
16257:
16251:
16227:
16221:
16210:. Retrieved
16206:
16196:
16181:
16154:
16145:
16071:
16067:
16063:
16059:
16055:
16047:
16043:
16035:
16031:
16021:
16001:
15992:
15980:
15975:
15967:
15965:
15962:Applications
15955:
15951:
15941:
15589:
15264:
15105:
14963:
14795:
14582:Subtracting
14581:
14381:
14332:
14330:
14207:is given by
14204:
14200:
14196:
14192:
14190:
13550:
13003:
12999:
12996:
12978:
12974:
12958:
12950:
12946:
12920:
12917:
12911:
12724:
12720:
12717:
12580:
12577:
12397:
12391:
12388:
12377:
12373:
12200:
12102:
12099:
12025:
12023:If a matrix
12022:
12010:
12006:
11995:
11991:
11976:
11970:
11390:
11384:
11316:
11305:
11006:
11003:
10818:
10812:
10802:
10799:
10792:
10781:
10471:
10465:
10462:
10447:
10445:
10434:
9759:
9753:
9751:
9740:
9321:
9318:
9314:
9308:
9302:
9296:
9290:
9284:
9277:
9275:
9269:
9263:
9256:
9241:
9235:
9225:
9222:
9218:
9212:
9206:
9203:
9199:
9193:
9187:
9177:
9171:
9165:
9159:
9156:
9145:
8728:
8726:
8400:
8394:
8391:
8246:
8243:
8239:
8236:
8100:
8094:
8090:
7989:
7980:
7971:
7964:
7957:
7954:
7552:The general
7551:
7364:
7298:as follows:
7290:
7287:
6864:
6858:
6848:
6522:
6412:
6401:
6397:
6393:
6154:
6152:
6137:
6127:
6121:
6109:
6104:
5931:
5615:
5605:
5588:
5582:denotes the
5577:
5571:
5558:
5555:
5459:
5456:
5365:
5311:, therefore
5303:
5297:
5241:
5235:
5194:
5190:
5183:
5180:
5098:
5092:
5089:
5071:
4765:
4637:
4626:
4619:
4612:
4311:
4304:
4297:
4290:
4265:
4255:
4163:
4152:
4146:
4136:
4130:
4045:
3844:
3719:
3643:
3463:
3301:
3203:This yields
3086:
2993:
2912:
2898:
2859:
2765:
2699:
2614:
2608:
2604:
2601:
2527:
2524:
2508:
2462:
2431:measure zero
2367:
2321:, then also
2317:
2311:
2306:
2304:
2258:
2166:
2163:
2161:as follows:
2157:
2151:
2149:of a matrix
2144:
2129:
2125:
2119:
2115:
2109:
2106:
2100:
2090:
1942:
1902:
1896:
1892:
1888:
1882:
1872:
1870:of a matrix
1866:
1863:
1471:
1465:
1367:
1357:
1146:
1143:
1127:
1118:
1079:
1053:
1031:
1028:is nonzero:
1024:
1007:
1005:The rows of
990:
969:
963:
959:
953:
947:
938:
934:
923:
915:
911:
900:
885:
876:
872:
852:
843:
839:
819:
811:
799:
793:
784:
779:
775:
771:
765:
759:
713:be a square
709:
706:
689:
684:
656:
652:
649:
614:
607:
603:
599:
593:
579:
575:
569:
563:
556:
552:
548:
542:
536:
530:
524:
518:
514:
508:
506:is equal to
502:
484:
476:left inverse
470:
466:
429:
425:
421:
415:
410:
405:
399:
392:
391:
384:
378:
361:denotes the
356:
352:
292:
279:
275:
271:
267:
262:
245:
230:
212:
203:
187:
157:
148:
138:
131:
124:
117:
105:
93:Please help
88:verification
85:
61:
54:
48:
47:Please help
44:
18150:Hamiltonian
18074:Biadjacency
18010:Correlation
17926:Householder
17876:Commutation
17613:Vandermonde
17608:Tridiagonal
17543:Permutation
17533:Nonnegative
17518:Matrix unit
17398:Bisymmetric
17196:Multivector
17161:Determinant
17118:Dot product
16963:Linear span
16012:ray casting
16008:3D graphics
15590:Therefore,
10808:zero matrix
9764:) leads to
6849:(where the
6117:determinant
5205:eigenvector
4617:is size of
3753:), such as
2700:The matrix
2435:determinant
2018:of any two
1878:orthonormal
1371:is a vector
1125:The matrix
1020:determinant
650:The set of
641:commutative
450:probability
442:number line
438:determinant
272:nonsingular
206:August 2021
18278:Categories
18050:Transition
18045:Stochastic
18015:Covariance
17997:statistics
17976:Symplectic
17971:Similarity
17800:Unimodular
17795:Orthogonal
17780:Involutory
17775:Invertible
17770:Projection
17766:Idempotent
17708:Convergent
17603:Triangular
17553:Polynomial
17498:Hessenberg
17468:Equivalent
17463:Elementary
17443:Copositive
17433:Conference
17393:Bidiagonal
17230:Direct sum
17065:Invertible
16968:Linear map
16212:2020-09-08
16137:References
9244:Hans Boltz
6408:reciprocal
5457:If matrix
5090:If matrix
4635:of matrix
4257:Victor Pan
4046:To obtain
2506:matrices.
2451:almost all
2443:polynomial
2113:such that
2086:orthogonal
1049:eigenvalue
983:surjective
769:such that
698:Properties
680:, denoted
676:of degree
597:such that
546:such that
430:degenerate
278:or rarely
268:invertible
266:is called
121:newspapers
50:improve it
18231:Wronskian
18155:Irregular
18145:Gell-Mann
18094:Laplacian
18089:Incidence
18069:Adjacency
18040:Precision
18005:Centering
17911:Generator
17881:Confusion
17866:Circulant
17846:Augmented
17805:Unipotent
17785:Nilpotent
17761:Congruent
17738:Stieltjes
17713:Defective
17703:Companion
17674:Redheffer
17593:Symmetric
17588:Sylvester
17563:Signature
17493:Hermitian
17473:Frobenius
17383:Arrowhead
17363:Alternant
17260:Numerical
17023:Transpose
16810:June 2015
16794:excessive
16680:EMS Press
16327:120467300
15905:ε
15879:−
15864:−
15844:−
15817:∑
15813:ε
15810:−
15802:−
15780:−
15767:ε
15736:ε
15713:−
15690:−
15660:∑
15656:ε
15616:ε
15560:−
15545:−
15504:−
15477:∑
15473:−
15445:−
15415:−
15371:−
15341:∑
15237:ε
15166:∑
15162:ε
15131:ε
15014:∑
14936:ε
14914:−
14894:−
14884:ε
14881:−
14873:−
14855:−
14841:ε
14804:ε
14773:−
14732:−
14722:−
14698:−
14655:−
14597:−
14496:−
14460:−
14413:−
14350:−
14308:−
14267:−
14257:−
14233:−
14128:∧
14125:⋯
14122:∧
14107:∧
14013:δ
13985:⋅
13940:⋅
13903:⋅
13865:−
13823:δ
13772:δ
13744:⋅
13716:−
13566:−
13531:−
13511:∧
13508:⋯
13505:∧
13487:∧
13469:⋅
13451:∧
13448:⋯
13445:∧
13429:∧
13426:⋯
13423:∧
13400:−
13386:−
13294:δ
13275:⋅
13076:≤
13064:≤
12997:Given an
12873:−
12857:−
12842:
12823:−
12805:−
12789:−
12776:−
12768:−
12750:−
12692:−
12663:−
12647:−
12629:∞
12614:∑
12602:−
12541:−
12526:−
12510:∞
12507:→
12449:−
12439:−
12423:∞
12420:→
12331:−
12302:−
12284:∏
12262:−
12246:−
12221:∑
12168:−
12155:∞
12140:∑
12128:−
12064:−
12051:∞
12048:→
11902:−
11862:−
11749:−
11729:−
11702:−
11621:−
11601:−
11571:−
11510:−
11429:−
11353:−
11338:−
11273:−
11256:−
11236:−
11226:−
11214:−
11187:−
11177:−
11167:−
11147:−
11120:−
11102:−
11076:−
10973:−
10956:−
10938:−
10928:−
10909:−
10883:−
10742:−
10727:−
10715:−
10700:−
10669:−
10650:−
10635:−
10598:−
10579:−
10564:−
10534:−
10452:) is the
10409:−
10390:−
10377:−
10351:−
10333:−
10314:−
10301:−
10277:−
10259:−
10240:−
10222:−
10203:−
10190:−
10164:−
10145:−
10132:−
10108:−
10089:−
10070:−
10057:−
10033:−
10011:−
9993:−
9974:−
9961:−
9938:−
9920:−
9901:−
9888:−
9864:−
9846:−
9824:−
9805:−
9792:−
9708:−
9690:−
9671:−
9658:−
9634:−
9616:−
9598:−
9579:−
9566:−
9542:−
9532:−
9520:−
9502:−
9483:−
9470:−
9456:−
9446:−
9427:−
9414:−
9384:−
9113:−
9094:−
9081:−
9060:−
9042:−
9023:−
9010:−
8996:−
8984:−
8965:−
8952:−
8928:−
8918:−
8908:−
8890:−
8871:−
8858:−
8834:−
8816:−
8790:−
8687:−
8679:
8635:
8629:−
8611:
8582:−
8551:
8521:
8510:
8501:−
8483:
8423:−
8354:×
8336:⋅
8303:×
8285:⋅
8207:×
8189:⋅
8127:−
8058:×
8040:⋅
7905:×
7857:×
7809:×
7750:−
7509:
7498:−
7467:
7461:−
7443:
7383:−
7257:−
7214:−
7202:−
7168:−
7126:−
7114:−
7080:−
7037:−
7025:−
6992:−
6949:−
6937:−
6903:−
6624:−
6548:−
6490:−
6472:
6431:−
6359:−
6349:−
6319:−
6284:−
6274:−
6228:−
6179:−
6143:transpose
5955:−
5892:⋯
5851:⋮
5846:⋱
5841:⋮
5836:⋮
5812:⋯
5760:⋯
5640:−
5620:recursive
5533:−
5518:−
5508:∗
5485:−
5420:λ
5390:−
5386:Λ
5326:−
5274:λ
5258:Λ
5158:−
5143:−
5138:Λ
5120:−
5046:−
5032:…
4995:−
4972:−
4956:−
4942:−
4928:−
4898:∑
4863:−
4815:
4800:−
4791:−
4748:−
4721:−
4703:∑
4667:≥
4561:
4494:−
4480:−
4462:∏
4451:−
4437:…
4407:∑
4386:−
4368:∑
4333:−
4261:John Reif
4215:−
4084:⋯
4076:−
4023:−
4005:−
3965:−
3918:⋯
3910:−
3860:−
3791:⋯
3783:−
3659:−
3538:→
3488:→
3475:−
3405:−
3397:−
3384:−
3340:→
3323:−
3219:−
3175:→
3049:−
3010:−
2966:−
2937:−
2831:−
2793:−
2741:−
2671:−
2642:−
2400:×
2309:matrices
2225:
2188:−
2057:δ
1996:≤
1984:≤
1841:−
1814:−
1775:−
1755:−
1740:⋯
1732:−
1724:−
1706:−
1683:−
1658:−
1645:⋯
1575:…
1532:−
1517:−
1499:−
1463:matrices
1424:−
1403:−
1338:−
1265:−
1250:−
1234:−
1184:−
1171:−
1099:λ
1096:−
1065:λ
979:injective
904:has full
808:transpose
567:has rank
296:such that
56:talk page
18289:Matrices
18059:Used in
17995:Used in
17956:Rotation
17931:Jacobian
17891:Distance
17871:Cofactor
17856:Carleman
17836:Adjugate
17820:Weighing
17753:inverses
17749:products
17718:Definite
17649:Identity
17639:Exchange
17632:Constant
17598:Toeplitz
17483:Hadamard
17453:Diagonal
17306:Category
17245:Subspace
17240:Quotient
17191:Bivector
17105:Bilinear
17047:Matrices
16922:Glossary
16844:Archived
16603:(2008),
16560:14 April
16078:See also
15958:matrix.
13814:, where
13099:vectors
12930:rational
9248:geodetic
5932:so that
4272:homotopy
4268:families
2521:Examples
2476:open set
2428:Lebesgue
2424:null set
2147:adjugate
629:addition
522:), then
496:and the
426:singular
18160:Overlap
18125:Density
18084:Edmonds
17961:Seifert
17921:Hessian
17886:Coxeter
17810:Unitary
17728:Hurwitz
17659:Of ones
17644:Hilbert
17578:Skyline
17523:Metzler
17513:Logical
17508:Integer
17418:Boolean
17350:classes
16917:Outline
16848:YouTube
16788:Please
16780:use of
16682:, 2001
16523:Bibcode
16464:Ran Raz
16307:Bibcode
16054:of the
13846:is the
13091:, with
12926:integer
10806:is the
10793:By the
9758:) and (
6406:is the
6131:is the
6115:is the
5562:is the
5245:is the
4631:is the
3880:we get
2482:of all
2478:in the
2420:
2382:
2364:Density
994:form a
860:to the
827:to the
752:
730:
668:form a
621:complex
591:matrix
540:matrix
416:Over a
393:inverse
280:regular
192:Please
135:scholar
18079:Degree
18020:Design
17951:Random
17941:Payoff
17936:Moment
17861:Cartan
17851:Bézout
17790:Normal
17664:Pascal
17654:Lehmer
17583:Sparse
17503:Hollow
17488:Hankel
17423:Cauchy
17348:Matrix
17201:Tensor
17013:Kernel
16943:Vector
16938:Scalar
16733:
16712:
16615:
16583:
16492:
16431:
16406:
16350:
16325:
16238:
16169:
15106:then,
13490:
12892:
12700:
12495:
12492:
12483:
12480:
11317:where
9157:where
7987:, and
7703:, and
6851:scalar
6135:, and
6113:|
6107:|
6105:where
5575:, and
5556:where
5239:, and
5181:where
4623:, and
4613:where
3644:Thus,
2607:− 1 ≠
2492:closed
2378:subset
2348:
2290:
981:" or "
929:kernel
672:, the
635:(i.e.
561:. If
448:, the
350:where
270:(also
137:
130:
123:
116:
108:
18140:Gamma
18104:Tutte
17966:Shear
17679:Shift
17669:Pauli
17618:Walsh
17528:Moore
17408:Block
17070:Minor
17055:Block
16993:Basis
16753:(PDF)
16323:S2CID
12941:-adic
12578:then
12100:then
9229:(the
7554:3 × 3
6528:3 × 3
6159:2 × 2
5616:small
4633:trace
4134:with
2473:dense
2422:is a
996:basis
937:) = {
910:rank
723:field
670:group
637:rings
482:. If
418:field
250:, an
142:JSTOR
128:books
17946:Pick
17916:Gram
17684:Zero
17388:Band
17225:Dual
17080:Rank
16731:ISBN
16710:ISBN
16613:ISBN
16581:ISBN
16562:2018
16490:ISBN
16429:ISBN
16404:ISBN
16348:ISBN
16236:ISBN
16167:ISBN
16024:MIMO
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2467:-by-
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11870:=
11865:1
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11414:=
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11330:=
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7786:[
7778:)
7774:A
7770:(
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7758:=
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7433:[
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7411:)
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7350:.
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7341:+
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7316:A
7312:(
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6930:=
6923:D
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6632:=
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5393:1
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4312:A
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4298:A
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4232:X
4228:A
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4219:X
4210:k
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4112:I
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4101:E
4094:2
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4079:1
4073:n
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4031:,
4026:1
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4013:=
4008:1
4000:A
3994:I
3973:.
3968:1
3960:A
3954:I
3950:=
3946:I
3940:1
3935:E
3928:2
3923:E
3913:1
3907:n
3902:E
3895:n
3890:E
3868:,
3863:1
3855:A
3831:.
3827:I
3823:=
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3813:1
3808:E
3801:2
3796:E
3786:1
3780:n
3775:E
3768:n
3763:E
3739:n
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3724:(
3706:.
3701:)
3695:2
3690:2
3683:3
3678:2
3672:(
3667:=
3662:1
3654:A
3630:.
3626:)
3619:2
3614:2
3609:1
3604:0
3597:3
3592:2
3587:0
3582:1
3575:(
3554:.
3551:)
3546:2
3542:R
3533:2
3529:R
3524:2
3521:(
3501:)
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3483:1
3479:R
3472:(
3450:.
3446:)
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3353:)
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3288:.
3284:)
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3246:0
3241:1
3233:2
3230:3
3222:1
3212:(
3191:.
3188:)
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3162:+
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3149:(
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3096:R
3073:.
3069:)
3062:1
3057:0
3052:1
3044:1
3037:0
3032:1
3024:2
3021:3
3013:1
3003:(
2980:.
2975:)
2969:1
2961:1
2951:2
2948:3
2940:1
2931:(
2926:=
2922:A
2869:C
2845:.
2840:)
2834:1
2823:3
2820:2
2807:2
2804:3
2796:1
2787:(
2782:=
2778:C
2749:2
2746:1
2738:=
2734:B
2709:B
2685:.
2680:)
2674:1
2666:1
2656:2
2653:3
2645:1
2636:(
2631:=
2627:B
2609:n
2605:n
2587:.
2582:)
2576:4
2571:2
2564:4
2559:2
2553:(
2548:=
2544:A
2528:n
2504:n
2500:n
2488:n
2484:n
2469:n
2465:n
2458:n
2454:n
2408:,
2403:n
2397:n
2392:R
2374:n
2370:n
2344:I
2340:=
2336:A
2333:B
2318:B
2312:A
2286:I
2282:=
2278:B
2275:A
2239:.
2236:)
2232:A
2228:(
2216:)
2212:A
2208:(
2201:1
2196:=
2191:1
2183:A
2167:A
2158:A
2152:A
2130:I
2126:A
2120:A
2116:A
2110:A
2101:V
2091:U
2072:.
2067:j
2064:,
2061:i
2053:=
2048:j
2044:u
2037:T
2031:i
2027:v
2002:.
1999:n
1993:j
1990:,
1987:i
1981:1
1959:j
1955:u
1943:U
1925:T
1919:i
1915:v
1903:V
1897:I
1883:U
1873:U
1867:V
1849:.
1844:1
1837:)
1832:A
1825:(
1822:=
1817:1
1809:A
1783:.
1778:1
1770:1
1765:A
1758:1
1750:2
1745:A
1735:1
1727:1
1721:k
1716:A
1709:1
1701:k
1696:A
1691:=
1686:1
1679:)
1673:k
1668:A
1661:1
1655:k
1650:A
1640:2
1635:A
1628:1
1623:A
1618:(
1608:n
1604:n
1588:k
1583:A
1578:,
1572:,
1567:1
1562:A
1540:.
1535:1
1527:A
1520:1
1512:B
1507:=
1502:1
1495:)
1490:B
1487:A
1483:(
1472:B
1466:A
1461:n
1457:n
1438:T
1433:)
1427:1
1419:A
1414:(
1411:=
1406:1
1399:)
1392:T
1386:A
1381:(
1368:x
1358:A
1341:1
1333:A
1326:+
1321:x
1316:=
1311:+
1307:)
1302:x
1299:A
1295:(
1284:k
1268:1
1260:A
1253:1
1246:k
1242:=
1237:1
1230:)
1225:A
1221:k
1218:(
1196:A
1192:=
1187:1
1180:)
1174:1
1166:A
1161:(
1147:A
1135:.
1128:A
1119:I
1103:I
1092:A
1080:A
1054:A
1032:A
1025:A
1013:K
1008:A
1000:K
991:A
975:K
970:b
964:b
948:x
939:0
935:A
924:A
919:.
916:n
912:A
901:A
896:.
891:n
886:A
881:.
877:n
873:I
866:n
862:n
853:A
848:.
844:n
840:I
833:n
829:n
820:A
812:A
794:x
780:n
776:I
766:B
760:A
739:R
726:K
719:n
715:n
710:A
692:)
690:R
688:(
685:n
678:n
666:R
657:n
653:n
608:m
604:I
594:B
589:m
585:n
580:n
576:m
573:(
570:m
564:A
557:n
553:I
543:B
537:m
531:n
525:A
519:m
515:n
509:n
503:A
494:n
490:m
485:A
471:n
467:m
462:n
458:m
406:A
400:A
385:A
379:B
367:n
363:n
357:n
353:I
338:,
333:n
328:I
323:=
319:A
316:B
312:=
308:B
305:A
293:B
288:n
284:n
263:A
256:n
252:n
237:)
231:(
219:)
213:(
208:)
204:(
190:.
164:)
158:(
153:)
149:(
139:·
132:·
125:·
118:·
91:.
66:)
62:(
20:)
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