4366:
2709:
4630:
2692:
2581:
2818:
This leads to a much more general definition of the transpose that works on every linear map, even when linear maps cannot be represented by matrices (such as in the case of infinite dimensional vector spaces). In the finite dimensional case, the matrix representing the transpose of a linear map is
31:
1163:
1343:
1524:
1039:
1845:
937:
2687:{\displaystyle \left(\mathbf {A} \mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }=\mathbf {A} \mathbf {A} ^{\operatorname {T} }.}
1416:
1249:
1739:
1659:
851:
1047:
704:
632:
3608:
1268:
776:
555:
494:
1459:
945:
2742:
However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in
2358:
2006:
1952:
1767:
864:
338:
1691:
154:
2380:
2329:
2303:
2194:
2172:
2060:
2038:
1977:
1919:
1358:
2086:
1199:
3738:
of a map between such spaces is defined similarly, and the matrix of the
Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal.
42:
can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.
2746:, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a
2278:
2254:
2234:
2214:
2150:
2126:
2106:
1897:
1604:
799:
1158:{\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}}
659:
590:
2739:, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
1338:{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.}
3672:
with respect to their bilinear forms. In this context, many authors however, use the term transpose to refer to the adjoint as defined here.
3509:
731:
517:
459:
3885:
5023:
4224:
1519:{\displaystyle \left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.}
1034:{\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3\\2&4\end{bmatrix}}}
1849:
The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.
4557:
4615:
1869:
5172:
4150:
4120:
4094:
4073:
4041:
3844:
3715:
with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps
97:
This article assumes that matrices are taken over a commutative ring. These results may not hold in the non-commutative case.
2757:
Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an
2012:, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties.
1420:
The transpose of a scalar is the same scalar. Together with the preceding property, this implies that the transpose is a
1840:{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\operatorname {T} }.}
5207:
4886:
3052:
The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint (
2334:
1982:
1928:
58:
is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix
4666:
4179:
3951:
2815:, the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps.
5088:
4605:
2937:
2750:
algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing
4033:
1528:
The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so
4567:
4503:
3761:
2812:
932:{\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\operatorname {T} }=\,{\begin{bmatrix}1\\2\end{bmatrix}}}
398:. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as
24:
4939:
4871:
4112:
2790:
2703:
4196:
267:
4964:
4345:
4217:
1759:
5202:
4450:
4300:
3766:
2743:
2713:
5013:
4833:
4355:
4249:
5315:
4685:
4595:
4244:
2832:
2806:
1873:
20:
5167:
4019:
132:
5269:
5187:
5141:
4848:
4587:
4470:
3064:
1411:{\displaystyle \left(c\mathbf {A} \right)^{\operatorname {T} }=c\mathbf {A} ^{\operatorname {T} }.}
1254:
404:. An advantage of this notation is that no parentheses are needed when exponents are involved: as
184:
2363:
2312:
2286:
2177:
2155:
2043:
2021:
1960:
1902:
5320:
5239:
4926:
4843:
4813:
4633:
4562:
4340:
4210:
3834:
1244:{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .}
5197:
5053:
5008:
4397:
4330:
4320:
3665:
3197:
2820:
2747:
2065:
430:
4169:
4084:
5279:
5234:
4714:
4659:
4412:
4407:
4402:
4335:
4280:
3812:
3060:
2420:
1734:{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} .}
643:
501:
5254:
5182:
5068:
4934:
4896:
4828:
4422:
4387:
4374:
4265:
4138:
2852:
2844:
2724:
1190:
55:
8:
5131:
4954:
4944:
4793:
4778:
4734:
4600:
4480:
4455:
4305:
3756:
575:
5264:
5121:
4974:
4788:
4724:
4310:
4104:
3410:
2728:
2463:
2306:
2263:
2239:
2219:
2199:
2135:
2111:
2091:
1882:
1654:{\displaystyle \det \left(\mathbf {A} ^{\operatorname {T} }\right)=\det(\mathbf {A} ).}
781:
A square complex matrix whose transpose is equal to its conjugate inverse is called a
5310:
5259:
5028:
5003:
4818:
4729:
4709:
4508:
4465:
4392:
4285:
4175:
4156:
4146:
4126:
4116:
4090:
4069:
4047:
4037:
3947:
3840:
3735:
3731:
3395:
2789:
of the data elements that is non-trivial to implement in-place. Therefore, efficient
2257:
2009:
1922:
715:
641:
matrix whose transpose is equal to the negation of its complex conjugate is called a
565:
86:
3860:
5274:
4949:
4916:
4901:
4783:
4652:
4513:
4417:
4270:
4115:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
4015:
3708:
2794:
570:
443:
846:{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} ^{-1}}}.}
5244:
5192:
5136:
5116:
5018:
4906:
4773:
4744:
4572:
4365:
4325:
4315:
3751:
3747:
3049:). This definition also applies unchanged to left modules and to vector spaces.
3023:
2950:
2751:
2129:
1348:
82:
5284:
5249:
5146:
4979:
4969:
4959:
4881:
4853:
4838:
4823:
4739:
4577:
4498:
4233:
3936:
3913:
3890:
3669:
2770:
2732:
1258:
783:
710:
638:
564:
matrix whose transpose is equal to the matrix with every entry replaced by its
561:
78:
47:
5229:
2819:
the transpose of the matrix representing the linear map, independently of the
699:{\displaystyle \mathbf {A} ^{\operatorname {T} }=-{\overline {\mathbf {A} }}.}
73:
The transpose of a matrix was introduced in 1858 by the
British mathematician
5304:
5221:
5126:
5038:
4911:
4610:
4533:
4493:
4460:
4440:
4160:
4130:
3413:
3227:
2708:
2473:
627:{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} }}.}
193:
74:
4051:
5289:
5093:
5078:
5043:
4891:
4876:
4543:
4432:
4382:
4275:
3787:
3664:. The matrix of the adjoint of a map is the transposed matrix only if the
3465:
3370:
1425:
2797:, starting in the late 1950s, and several algorithms have been developed.
5177:
5151:
5073:
4762:
4701:
4523:
4488:
4445:
4290:
4061:
3944:
3823: : 17–37. The transpose (or "transposition") is defined on page 31.
3635:
3383:
2786:
1872:
are equal to the eigenvalues of its transpose, since they share the same
1673:
1664:
3291:. By defining the transpose of this bilinear form as the bilinear form
5058:
4552:
4295:
3603:{\displaystyle B_{X}{\big (}x,g(y){\big )}=B_{Y}{\big (}u(x),y{\big )}}
3461:
3215:
2885:
1421:
2697:
1857:
is sometimes used to represent either of these equivalent expressions.
5033:
2727:
by simply accessing the same data in a different order. For example,
499:
A square matrix whose transpose is equal to its negative is called a
771:{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} ^{-1}.}
425:
In this article this confusion is avoided by never using the symbol
5063:
5048:
4518:
3734:(conjugate-linear in one argument) instead of bilinear forms. The
3660:, resulting in an isomorphism between the transpose and adjoint of
2766:
2720:
1667:
of a square matrix is the same as the determinant of its transpose.
30:
4757:
4719:
4202:
2811:
As the main use of matrices is to represent linear maps between
550:{\displaystyle \mathbf {A} ^{\operatorname {T} }=-\mathbf {A} .}
5083:
4675:
4528:
3214:
describes the transpose of that linear map with respect to the
2800:
2500:, so the entry corresponds to the inner product of two rows of
489:{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} .}
441:
A square matrix whose transpose is equal to itself is called a
2382:
is a single Jordan block, which is a straightforward exercise.
2015:
A proof of this property uses the following two observations.
3441:, which is closely related to the transpose, may be defined:
436:
3956:
2736:
4644:
4036:. Berlin New York: Springer Science & Business Media.
2793:
has been the subject of numerous research publications in
180:, may be constructed by any one of the following methods:
3817:
2723:, one can often avoid explicitly transposing a matrix in
1688:
can be computed as the single entry of the matrix product
16:
Matrix operation which flips a matrix over its diagonal
3886:"What is the best symbol for vector/matrix transpose?"
2513:
is the entry of the product, it is obtained from rows
1114:
1057:
1000:
955:
908:
874:
3992:
3512:
2773:
additional storage or at most storage much less than
2584:
2366:
2360:. This further reduces to proving the same fact when
2337:
2315:
2289:
2266:
2242:
2222:
2202:
2180:
2158:
2138:
2114:
2094:
2068:
2046:
2024:
1985:
1963:
1931:
1905:
1885:
1770:
1694:
1607:
1462:
1361:
1271:
1202:
1050:
948:
867:
802:
734:
662:
593:
520:
462:
270:
196:(which runs from top-left to bottom-right) to obtain
135:
4143:
Topological Vector Spaces, Distributions and
Kernels
2575:
results from the fact that it is its own transpose:
19:
This article is about the transpose of matrices and
2698:
Implementation of matrix transposition on computers
2423:with these two matrices gives two square matrices:
3730:Over a complex vector space, one often works with
3602:
2686:
2374:
2352:
2323:
2305:is a matrix over an algebraically closed field in
2297:
2272:
2248:
2228:
2208:
2188:
2166:
2144:
2120:
2100:
2080:
2054:
2032:
2000:
1971:
1946:
1913:
1891:
1839:
1733:
1653:
1518:
1410:
1337:
1243:
1157:
1033:
931:
845:
770:
698:
626:
549:
488:
332:
148:
2353:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
2001:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
1947:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
5302:
1634:
1608:
709:A square matrix whose transpose is equal to its
3221:
64:by producing another matrix, often denoted by
4660:
4218:
4103:
3962:
3832:
3595:
3570:
3550:
3525:
574:(equivalent to the matrix being equal to its
2826:
2801:Transposes of linear maps and bilinear forms
1253:The operation of taking the transpose is an
568:(denoted here with an overline) is called a
4089:. San José: Solar Crest. pp. 122–132.
3861:"Transpose of a Matrix Product (ProofWiki)"
3836:Introduction to Linear Algebra, 2nd edition
3826:
4667:
4653:
4225:
4211:
3727:for which the adjoint equals the inverse.
3675:The adjoint allows us to consider whether
437:Matrix definitions involving transposition
902:
5024:Covariance and contravariance of vectors
4167:
4082:
4014:
3986:
2707:
92:
29:
3196:describes a linear map with respect to
2953:characterizes the algebraic adjoint of
2557:) is symmetric. Similarly, the product
2534:is also obtained from these rows, thus
5303:
4616:Comparison of linear algebra libraries
4137:
4060:
3998:
3974:
333:{\displaystyle \left_{ij}=\left_{ji}.}
4648:
4206:
4145:. Mineola, N.Y.: Dover Publications.
3911:
2419:is its transpose, then the result of
4174:. Mineola: Dover. pp. 126–132.
4171:Introduction to Matrices and Vectors
3813:"A memoir on the theory of matrices"
85:R, the transpose corresponds to the
3941:Linear Algebra and its Applications
3232:Every linear map to the dual space
13:
4887:Tensors in curvilinear coordinates
4232:
4008:
3785:
3707:. In particular, this allows the
2676:
2656:
2644:
2634:
2614:
2603:
2462:. Furthermore, these products are
2345:
2236:. In particular this applies when
1993:
1939:
1829:
1783:
1718:
1622:
1508:
1496:
1481:
1400:
1382:
1327:
1312:
1297:
1225:
1215:
1101:
987:
894:
810:
742:
670:
601:
528:
470:
283:
14:
5332:
4189:
2569:A quick proof of the symmetry of
2309:with respect to some basis, then
4629:
4628:
4606:Basic Linear Algebra Subprograms
4364:
4066:Finite dimensional vector spaces
3019:⟨•, •⟩
2813:finite-dimensional vector spaces
2671:
2665:
2651:
2629:
2598:
2592:
2368:
2340:
2317:
2291:
2182:
2160:
2048:
2026:
1988:
1965:
1934:
1907:
1811:
1778:
1724:
1713:
1704:
1696:
1641:
1617:
1503:
1491:
1472:
1469:
1395:
1372:
1322:
1307:
1287:
1279:
1234:
1210:
822:
805:
752:
737:
684:
665:
612:
596:
540:
523:
479:
465:
382:In the case of square matrices,
310:
278:
4504:Seven-dimensional cross product
3980:
3634:These bilinear forms define an
3968:
3943:4th edition, page 51, Thomson
3930:
3905:
3878:
3853:
3833:T.A. Whitelaw (1 April 1991).
3805:
3779:
3584:
3578:
3545:
3539:
2785:, this involves a complicated
2088:matrices over some base field
1645:
1637:
149:{\displaystyle A^{\intercal }}
25:Transposition (disambiguation)
1:
4940:Exterior covariant derivative
4872:Tensor (intrinsic definition)
4195:Gilbert Strang (Spring 2010)
3772:
2791:in-place matrix transposition
2704:In-place matrix transposition
2466:. Indeed, the matrix product
2216:, then they are similar over
2196:are similar as matrices over
1868:is a square matrix, then its
1440:matrices to the space of the
1168:
100:
4965:Raising and lowering indices
4346:Eigenvalues and eigenvectors
4107:; Wolff, Manfred P. (1999).
3222:Transpose of a bilinear form
3053:
2924:. The resulting functional
2375:{\displaystyle \mathbf {A} }
2324:{\displaystyle \mathbf {A} }
2298:{\displaystyle \mathbf {A} }
2189:{\displaystyle \mathbf {B} }
2167:{\displaystyle \mathbf {A} }
2055:{\displaystyle \mathbf {B} }
2033:{\displaystyle \mathbf {A} }
1972:{\displaystyle \mathbf {A} }
1914:{\displaystyle \mathbf {A} }
1760:positive-semidefinite matrix
1749:has only real entries, then
835:
688:
616:
7:
5203:Gluon field strength tensor
4674:
4168:Schwartz, Jacob T. (2001).
4083:Maruskin, Jared M. (2012).
3788:"The transpose of a matrix"
3767:Projection (linear algebra)
3762:Moore–Penrose pseudoinverse
3741:
3090:are TVSs then a linear map
2714:row- and column-major order
2390:
856:
10:
5337:
5014:Cartan formalism (physics)
4834:Penrose graphical notation
3393:
3389:
3225:
3156:denote the restriction of
2830:
2804:
2701:
2550:, and the product matrix (
105:The transpose of a matrix
96:
70:(among other notations).
18:
5220:
5160:
5109:
5102:
4994:
4925:
4862:
4806:
4753:
4700:
4693:
4686:Glossary of tensor theory
4682:
4624:
4586:
4542:
4479:
4431:
4373:
4362:
4258:
4240:
4109:Topological Vector Spaces
3963:Schaefer & Wolff 1999
3437:, a concept known as the
3297:defined by the transpose
2833:Transpose of a linear map
2827:Transpose of a linear map
2807:Transpose of a linear map
2472:has entries that are the
2081:{\displaystyle n\times n}
1874:characteristic polynomial
5270:Gregorio Ricci-Curbastro
5142:Riemann curvature tensor
4849:Van der Waerden notation
4199:from MIT Open Courseware
4086:Essential Linear Algebra
4034:Éléments de mathématique
4026:Algèbre: Chapitres 1 à 3
3394:Not to be confused with
3246:defines a bilinear form
3065:topological vector space
5240:Elwin Bruno Christoffel
5173:Angular momentum tensor
4844:Tetrad (index notation)
4814:Abstract index notation
3750:, the transpose of the
3132:, in which case we let
2566:is a symmetric matrix.
1347:The transpose respects
392:th power of the matrix
5054:Levi-Civita connection
4331:Row and column vectors
4021:Algebra I Chapters 1-3
3604:
2748:fast Fourier transform
2716:
2688:
2376:
2354:
2325:
2299:
2274:
2250:
2230:
2210:
2190:
2168:
2146:
2122:
2102:
2082:
2056:
2034:
2002:
1973:
1948:
1915:
1893:
1841:
1735:
1676:of two column vectors
1655:
1520:
1412:
1339:
1245:
1159:
1035:
933:
847:
772:
700:
628:
551:
490:
334:
255:-th column element of
241:-th column element of
150:
43:
23:. For other uses, see
5280:Jan Arnoldus Schouten
5235:Augustin-Louis Cauchy
4715:Differential geometry
4336:Row and column spaces
4281:Scalar multiplication
3918:mathworld.wolfram.com
3811:Arthur Cayley (1858)
3605:
3401:If the vector spaces
3061:continuous dual space
2711:
2689:
2488:. But the columns of
2421:matrix multiplication
2377:
2355:
2326:
2300:
2275:
2251:
2231:
2211:
2191:
2169:
2147:
2123:
2103:
2083:
2057:
2035:
2003:
1974:
1949:
1916:
1894:
1842:
1736:
1656:
1521:
1413:
1340:
1246:
1160:
1036:
934:
848:
773:
701:
653:is skew-Hermitian if
644:skew-Hermitian matrix
629:
552:
511:is skew-symmetric if
502:skew-symmetric matrix
491:
335:
218:Write the columns of
151:
93:Transpose of a matrix
33:
5255:Carl Friedrich Gauss
5188:stress–energy tensor
5183:Cauchy stress tensor
4935:Covariant derivative
4897:Antisymmetric tensor
4829:Multi-index notation
4471:Gram–Schmidt process
4423:Gaussian elimination
3711:over a vector space
3510:
3264:, with the relation
2845:algebraic dual space
2582:
2364:
2335:
2313:
2287:
2264:
2240:
2220:
2200:
2178:
2156:
2136:
2112:
2092:
2066:
2044:
2022:
1983:
1961:
1929:
1903:
1883:
1768:
1692:
1605:
1460:
1359:
1269:
1200:
1048:
946:
865:
800:
732:
660:
591:
518:
460:
388:may also denote the
268:
133:
5132:Nonmetricity tensor
4987:(2nd-order tensors)
4955:Hodge star operator
4945:Exterior derivative
4794:Transport phenomena
4779:Continuum mechanics
4735:Multilinear algebra
4601:Numerical stability
4481:Multilinear algebra
4456:Inner product space
4306:Linear independence
4105:Schaefer, Helmut H.
3912:Weisstein, Eric W.
3757:Conjugate transpose
576:conjugate transpose
77:. In the case of a
5265:Tullio Levi-Civita
5208:Metric tensor (GR)
5122:Levi-Civita symbol
4975:Tensor contraction
4789:General relativity
4725:Euclidean geometry
4311:Linear combination
3732:sesquilinear forms
3600:
3409:have respectively
3208:, then the matrix
2729:software libraries
2717:
2684:
2464:symmetric matrices
2372:
2350:
2321:
2307:Jordan normal form
2295:
2270:
2246:
2226:
2206:
2186:
2164:
2142:
2118:
2098:
2078:
2052:
2030:
1998:
1969:
1957:This implies that
1944:
1911:
1899:, a square matrix
1889:
1837:
1731:
1651:
1516:
1408:
1335:
1241:
1155:
1149:
1094:
1031:
1025:
980:
929:
923:
887:
843:
768:
696:
624:
547:
486:
422:is not ambiguous.
330:
210:as the columns of
204:Write the rows of
146:
44:
5298:
5297:
5260:Hermann Grassmann
5216:
5215:
5168:Moment of inertia
5029:Differential form
5004:Affine connection
4819:Einstein notation
4802:
4801:
4730:Exterior calculus
4710:Coordinate system
4642:
4641:
4509:Geometric algebra
4466:Kronecker product
4301:Linear projection
4286:Vector projection
4152:978-0-486-45352-1
4122:978-1-4612-7155-0
4096:978-0-9850627-3-6
4075:978-0-387-90093-3
4043:978-3-540-64243-5
4016:Bourbaki, Nicolas
3846:978-0-7514-0159-2
3736:Hermitian adjoint
3396:Hermitian adjoint
3106:weakly continuous
3026:(i.e. defined by
2949:. The following
2890:algebraic adjoint
2482:with a column of
2273:{\displaystyle k}
2258:algebraic closure
2249:{\displaystyle L}
2229:{\displaystyle k}
2209:{\displaystyle L}
2145:{\displaystyle k}
2121:{\displaystyle L}
2101:{\displaystyle k}
2010:invariant factors
1892:{\displaystyle k}
838:
725:is orthogonal if
716:orthogonal matrix
691:
619:
566:complex conjugate
87:converse relation
5328:
5316:Abstract algebra
5275:Bernhard Riemann
5107:
5106:
4950:Exterior product
4917:Two-point tensor
4902:Symmetric tensor
4784:Electromagnetism
4698:
4697:
4669:
4662:
4655:
4646:
4645:
4632:
4631:
4514:Exterior algebra
4451:Hadamard product
4368:
4356:Linear equations
4227:
4220:
4213:
4204:
4203:
4185:
4164:
4139:Trèves, François
4134:
4100:
4078:
4055:
4031:
4002:
3996:
3990:
3984:
3978:
3972:
3966:
3960:
3954:
3934:
3928:
3927:
3925:
3924:
3909:
3903:
3902:
3900:
3898:
3882:
3876:
3875:
3873:
3871:
3857:
3851:
3850:
3830:
3824:
3809:
3803:
3802:
3800:
3798:
3783:
3726:
3725:
3714:
3709:orthogonal group
3706:
3705:
3690:
3689:
3663:
3659:
3653:
3647:
3641:
3629:
3619:
3609:
3607:
3606:
3601:
3599:
3598:
3574:
3573:
3567:
3566:
3554:
3553:
3529:
3528:
3522:
3521:
3502:
3501:
3486:
3478:
3474:
3470:
3459:
3458:
3436:
3425:
3408:
3404:
3381:
3368:
3364:
3337:
3310:
3296:
3290:
3263:
3245:
3213:
3207:
3203:
3195:
3186:
3178:
3172:
3171:
3170:
3161:
3155:
3154:
3153:
3145:
3144:
3131:
3130:
3129:
3121:
3120:
3103:
3089:
3085:
3081:
3080:
3079:
3070:
3048:
3038:
3021:
3020:
3012:
3002:
2992:
2991:
2975:
2956:
2948:
2944:
2934:
2923:
2909:
2883:
2869:
2865:
2861:
2857:
2850:
2842:
2795:computer science
2712:Illustration of
2693:
2691:
2690:
2685:
2680:
2679:
2674:
2668:
2660:
2659:
2654:
2648:
2647:
2642:
2638:
2637:
2632:
2618:
2617:
2612:
2608:
2607:
2606:
2601:
2595:
2574:
2565:
2556:
2549:
2533:
2526:
2520:
2516:
2512:
2505:
2499:
2494:are the rows of
2493:
2487:
2481:
2471:
2461:
2460:
2449:
2440:
2439:
2428:
2418:
2412:
2411:
2400:
2381:
2379:
2378:
2373:
2371:
2359:
2357:
2356:
2351:
2349:
2348:
2343:
2330:
2328:
2327:
2322:
2320:
2304:
2302:
2301:
2296:
2294:
2279:
2277:
2276:
2271:
2255:
2253:
2252:
2247:
2235:
2233:
2232:
2227:
2215:
2213:
2212:
2207:
2195:
2193:
2192:
2187:
2185:
2173:
2171:
2170:
2165:
2163:
2151:
2149:
2148:
2143:
2127:
2125:
2124:
2119:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2061:
2059:
2058:
2053:
2051:
2039:
2037:
2036:
2031:
2029:
2007:
2005:
2004:
1999:
1997:
1996:
1991:
1978:
1976:
1975:
1970:
1968:
1953:
1951:
1950:
1945:
1943:
1942:
1937:
1920:
1918:
1917:
1912:
1910:
1898:
1896:
1895:
1890:
1867:
1856:
1846:
1844:
1843:
1838:
1833:
1832:
1827:
1823:
1822:
1814:
1800:
1799:
1791:
1787:
1786:
1781:
1757:
1748:
1740:
1738:
1737:
1732:
1727:
1722:
1721:
1716:
1707:
1699:
1687:
1681:
1660:
1658:
1657:
1652:
1644:
1630:
1626:
1625:
1620:
1594:
1525:
1523:
1522:
1517:
1512:
1511:
1506:
1500:
1499:
1494:
1485:
1484:
1479:
1475:
1451:
1450:
1439:
1438:
1417:
1415:
1414:
1409:
1404:
1403:
1398:
1386:
1385:
1380:
1376:
1375:
1344:
1342:
1341:
1336:
1331:
1330:
1325:
1316:
1315:
1310:
1301:
1300:
1295:
1291:
1290:
1282:
1250:
1248:
1247:
1242:
1237:
1229:
1228:
1223:
1219:
1218:
1213:
1188:
1185:be matrices and
1184:
1178:
1164:
1162:
1161:
1156:
1154:
1153:
1105:
1104:
1099:
1098:
1040:
1038:
1037:
1032:
1030:
1029:
991:
990:
985:
984:
938:
936:
935:
930:
928:
927:
898:
897:
892:
891:
852:
850:
849:
844:
839:
834:
833:
825:
819:
814:
813:
808:
792:
777:
775:
774:
769:
764:
763:
755:
746:
745:
740:
724:
705:
703:
702:
697:
692:
687:
682:
674:
673:
668:
652:
633:
631:
630:
625:
620:
615:
610:
605:
604:
599:
584:is Hermitian if
583:
571:Hermitian matrix
556:
554:
553:
548:
543:
532:
531:
526:
510:
495:
493:
492:
487:
482:
474:
473:
468:
453:is symmetric if
452:
444:symmetric matrix
428:
421:
415:
403:
397:
391:
387:
378:
377:
366:
360:
359:
348:
339:
337:
336:
331:
326:
325:
317:
313:
300:
299:
291:
287:
286:
281:
260:
254:
250:
246:
240:
236:
229:
223:
215:
209:
201:
191:
179:
173:
167:
161:
155:
153:
152:
147:
145:
144:
128:
122:
116:
110:
69:
63:
21:linear operators
5336:
5335:
5331:
5330:
5329:
5327:
5326:
5325:
5301:
5300:
5299:
5294:
5245:Albert Einstein
5212:
5193:Einstein tensor
5156:
5137:Ricci curvature
5117:Kronecker delta
5103:Notable tensors
5098:
5019:Connection form
4996:
4990:
4921:
4907:Tensor operator
4864:
4858:
4798:
4774:Computer vision
4767:
4749:
4745:Tensor calculus
4689:
4678:
4673:
4643:
4638:
4620:
4582:
4538:
4475:
4427:
4369:
4360:
4326:Change of basis
4316:Multilinear map
4254:
4236:
4231:
4192:
4182:
4153:
4123:
4097:
4076:
4044:
4029:
4011:
4009:Further reading
4006:
4005:
3997:
3993:
3985:
3981:
3973:
3969:
3961:
3957:
3935:
3931:
3922:
3920:
3910:
3906:
3896:
3894:
3884:
3883:
3879:
3869:
3867:
3859:
3858:
3854:
3847:
3831:
3827:
3810:
3806:
3796:
3794:
3786:Nykamp, Duane.
3784:
3780:
3775:
3752:cofactor matrix
3748:Adjugate matrix
3744:
3717:
3716:
3712:
3693:
3692:
3677:
3676:
3661:
3655:
3649:
3643:
3639:
3621:
3611:
3594:
3593:
3569:
3568:
3562:
3558:
3549:
3548:
3524:
3523:
3517:
3513:
3511:
3508:
3507:
3489:
3488:
3484:
3476:
3472:
3468:
3446:
3445:
3435:
3427:
3424:
3416:
3406:
3402:
3399:
3392:
3373:
3369:is the natural
3366:
3339:
3338:, we find that
3312:
3298:
3292:
3265:
3247:
3233:
3230:
3224:
3209:
3205:
3201:
3191:
3184:
3174:
3168:
3167:
3163:
3157:
3151:
3150:
3142:
3141:
3133:
3127:
3126:
3118:
3117:
3109:
3108:if and only if
3091:
3087:
3083:
3077:
3076:
3072:
3068:
3028:
3027:
3024:natural pairing
3018:
3017:
3004:
2994:
2977:
2961:
2960:
2954:
2946:
2942:
2925:
2911:
2897:
2871:
2867:
2863:
2859:
2855:
2848:
2838:
2835:
2829:
2809:
2803:
2752:memory locality
2744:row-major order
2706:
2700:
2675:
2670:
2669:
2664:
2655:
2650:
2649:
2643:
2633:
2628:
2627:
2623:
2622:
2613:
2602:
2597:
2596:
2591:
2590:
2586:
2585:
2583:
2580:
2579:
2570:
2558:
2555:
2551:
2548:
2541:
2535:
2532:
2528:
2522:
2518:
2514:
2511:
2507:
2501:
2495:
2489:
2483:
2477:
2467:
2452:
2451:
2442:
2431:
2430:
2424:
2414:
2403:
2402:
2396:
2393:
2367:
2365:
2362:
2361:
2344:
2339:
2338:
2336:
2333:
2332:
2316:
2314:
2311:
2310:
2290:
2288:
2285:
2284:
2265:
2262:
2261:
2241:
2238:
2237:
2221:
2218:
2217:
2201:
2198:
2197:
2181:
2179:
2176:
2175:
2159:
2157:
2154:
2153:
2137:
2134:
2133:
2130:field extension
2113:
2110:
2109:
2093:
2090:
2089:
2067:
2064:
2063:
2047:
2045:
2042:
2041:
2025:
2023:
2020:
2019:
1992:
1987:
1986:
1984:
1981:
1980:
1964:
1962:
1959:
1958:
1938:
1933:
1932:
1930:
1927:
1926:
1906:
1904:
1901:
1900:
1884:
1881:
1880:
1879:Over any field
1863:
1852:
1850:
1828:
1815:
1810:
1809:
1805:
1804:
1792:
1782:
1777:
1776:
1772:
1771:
1769:
1766:
1765:
1750:
1744:
1723:
1717:
1712:
1711:
1703:
1695:
1693:
1690:
1689:
1683:
1677:
1640:
1621:
1616:
1615:
1611:
1606:
1603:
1602:
1593:
1587:
1580:
1571:
1562:
1554:
1544:
1538:
1531:
1507:
1502:
1501:
1495:
1490:
1489:
1480:
1468:
1464:
1463:
1461:
1458:
1457:
1442:
1441:
1430:
1429:
1399:
1394:
1393:
1381:
1371:
1367:
1363:
1362:
1360:
1357:
1356:
1326:
1321:
1320:
1311:
1306:
1305:
1296:
1286:
1278:
1277:
1273:
1272:
1270:
1267:
1266:
1233:
1224:
1214:
1209:
1208:
1204:
1203:
1201:
1198:
1197:
1186:
1180:
1174:
1171:
1148:
1147:
1142:
1137:
1131:
1130:
1125:
1120:
1110:
1109:
1100:
1093:
1092:
1087:
1081:
1080:
1075:
1069:
1068:
1063:
1053:
1052:
1051:
1049:
1046:
1045:
1024:
1023:
1018:
1012:
1011:
1006:
996:
995:
986:
979:
978:
973:
967:
966:
961:
951:
950:
949:
947:
944:
943:
922:
921:
915:
914:
904:
903:
893:
886:
885:
880:
870:
869:
868:
866:
863:
862:
859:
826:
821:
820:
818:
809:
804:
803:
801:
798:
797:
788:
756:
751:
750:
741:
736:
735:
733:
730:
729:
720:
683:
681:
669:
664:
663:
661:
658:
657:
648:
611:
609:
600:
595:
594:
592:
589:
588:
579:
539:
527:
522:
521:
519:
516:
515:
506:
478:
469:
464:
463:
461:
458:
457:
448:
439:
426:
417:
405:
399:
393:
389:
383:
369:
368:
362:
351:
350:
344:
318:
309:
305:
304:
292:
282:
277:
276:
272:
271:
269:
266:
265:
256:
252:
248:
242:
238:
234:
225:
224:as the rows of
219:
211:
205:
197:
187:
175:
169:
163:
157:
140:
136:
134:
131:
130:
124:
118:
112:
106:
103:
98:
95:
83:binary relation
81:representing a
65:
59:
28:
17:
12:
11:
5:
5334:
5324:
5323:
5321:Linear algebra
5318:
5313:
5296:
5295:
5293:
5292:
5287:
5285:Woldemar Voigt
5282:
5277:
5272:
5267:
5262:
5257:
5252:
5250:Leonhard Euler
5247:
5242:
5237:
5232:
5226:
5224:
5222:Mathematicians
5218:
5217:
5214:
5213:
5211:
5210:
5205:
5200:
5195:
5190:
5185:
5180:
5175:
5170:
5164:
5162:
5158:
5157:
5155:
5154:
5149:
5147:Torsion tensor
5144:
5139:
5134:
5129:
5124:
5119:
5113:
5111:
5104:
5100:
5099:
5097:
5096:
5091:
5086:
5081:
5076:
5071:
5066:
5061:
5056:
5051:
5046:
5041:
5036:
5031:
5026:
5021:
5016:
5011:
5006:
5000:
4998:
4992:
4991:
4989:
4988:
4982:
4980:Tensor product
4977:
4972:
4970:Symmetrization
4967:
4962:
4960:Lie derivative
4957:
4952:
4947:
4942:
4937:
4931:
4929:
4923:
4922:
4920:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4882:Tensor density
4879:
4874:
4868:
4866:
4860:
4859:
4857:
4856:
4854:Voigt notation
4851:
4846:
4841:
4839:Ricci calculus
4836:
4831:
4826:
4824:Index notation
4821:
4816:
4810:
4808:
4804:
4803:
4800:
4799:
4797:
4796:
4791:
4786:
4781:
4776:
4770:
4768:
4766:
4765:
4760:
4754:
4751:
4750:
4748:
4747:
4742:
4740:Tensor algebra
4737:
4732:
4727:
4722:
4720:Dyadic algebra
4717:
4712:
4706:
4704:
4695:
4691:
4690:
4683:
4680:
4679:
4672:
4671:
4664:
4657:
4649:
4640:
4639:
4637:
4636:
4625:
4622:
4621:
4619:
4618:
4613:
4608:
4603:
4598:
4596:Floating-point
4592:
4590:
4584:
4583:
4581:
4580:
4578:Tensor product
4575:
4570:
4565:
4563:Function space
4560:
4555:
4549:
4547:
4540:
4539:
4537:
4536:
4531:
4526:
4521:
4516:
4511:
4506:
4501:
4499:Triple product
4496:
4491:
4485:
4483:
4477:
4476:
4474:
4473:
4468:
4463:
4458:
4453:
4448:
4443:
4437:
4435:
4429:
4428:
4426:
4425:
4420:
4415:
4413:Transformation
4410:
4405:
4403:Multiplication
4400:
4395:
4390:
4385:
4379:
4377:
4371:
4370:
4363:
4361:
4359:
4358:
4353:
4348:
4343:
4338:
4333:
4328:
4323:
4318:
4313:
4308:
4303:
4298:
4293:
4288:
4283:
4278:
4273:
4268:
4262:
4260:
4259:Basic concepts
4256:
4255:
4253:
4252:
4247:
4241:
4238:
4237:
4234:Linear algebra
4230:
4229:
4222:
4215:
4207:
4201:
4200:
4197:Linear Algebra
4191:
4190:External links
4188:
4187:
4186:
4180:
4165:
4151:
4135:
4121:
4101:
4095:
4080:
4074:
4057:
4056:
4042:
4010:
4007:
4004:
4003:
4001:, p. 240.
3991:
3979:
3967:
3965:, p. 128.
3955:
3937:Gilbert Strang
3929:
3904:
3891:Stack Exchange
3877:
3852:
3845:
3825:
3804:
3777:
3776:
3774:
3771:
3770:
3769:
3764:
3759:
3754:
3743:
3740:
3648:, and between
3632:
3631:
3597:
3592:
3589:
3586:
3583:
3580:
3577:
3572:
3565:
3561:
3557:
3552:
3547:
3544:
3541:
3538:
3535:
3532:
3527:
3520:
3516:
3431:
3420:
3414:bilinear forms
3391:
3388:
3226:Main article:
3223:
3220:
3190:If the matrix
3179:is called the
3071:is denoted by
3014:
3013:
2935:is called the
2870:-modules. If
2831:Main article:
2828:
2825:
2802:
2799:
2733:linear algebra
2699:
2696:
2695:
2694:
2683:
2678:
2673:
2667:
2663:
2658:
2653:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2605:
2600:
2594:
2589:
2553:
2546:
2539:
2530:
2509:
2392:
2389:
2388:
2387:
2386:
2385:
2384:
2383:
2370:
2347:
2342:
2331:is similar to
2319:
2293:
2281:
2269:
2245:
2225:
2205:
2184:
2162:
2141:
2117:
2097:
2077:
2074:
2071:
2050:
2028:
2013:
2008:have the same
1995:
1990:
1967:
1941:
1936:
1909:
1888:
1877:
1860:
1859:
1858:
1836:
1831:
1826:
1821:
1818:
1813:
1808:
1803:
1798:
1795:
1790:
1785:
1780:
1775:
1763:
1741:
1730:
1726:
1720:
1715:
1710:
1706:
1702:
1698:
1670:
1669:
1668:
1650:
1647:
1643:
1639:
1636:
1633:
1629:
1624:
1619:
1614:
1610:
1600:
1599:
1598:
1597:
1596:
1591:
1585:
1575:
1567:
1563:) =
1558:
1549:
1542:
1536:
1515:
1510:
1505:
1498:
1493:
1488:
1483:
1478:
1474:
1471:
1467:
1455:
1454:
1453:
1407:
1402:
1397:
1392:
1389:
1384:
1379:
1374:
1370:
1366:
1354:
1353:
1352:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1285:
1281:
1276:
1264:
1263:
1262:
1240:
1236:
1232:
1227:
1222:
1217:
1212:
1207:
1170:
1167:
1166:
1165:
1152:
1146:
1143:
1141:
1138:
1136:
1133:
1132:
1129:
1126:
1124:
1121:
1119:
1116:
1115:
1113:
1108:
1103:
1097:
1091:
1088:
1086:
1083:
1082:
1079:
1076:
1074:
1071:
1070:
1067:
1064:
1062:
1059:
1058:
1056:
1042:
1041:
1028:
1022:
1019:
1017:
1014:
1013:
1010:
1007:
1005:
1002:
1001:
999:
994:
989:
983:
977:
974:
972:
969:
968:
965:
962:
960:
957:
956:
954:
940:
939:
926:
920:
917:
916:
913:
910:
909:
907:
901:
896:
890:
884:
881:
879:
876:
875:
873:
858:
855:
854:
853:
842:
837:
832:
829:
824:
817:
812:
807:
793:is unitary if
784:unitary matrix
779:
778:
767:
762:
759:
754:
749:
744:
739:
707:
706:
695:
690:
686:
680:
677:
672:
667:
635:
634:
623:
618:
614:
608:
603:
598:
558:
557:
546:
542:
538:
535:
530:
525:
497:
496:
485:
481:
477:
472:
467:
438:
435:
341:
340:
329:
324:
321:
316:
312:
308:
303:
298:
295:
290:
285:
280:
275:
233:Formally, the
231:
230:
216:
202:
143:
139:
102:
99:
94:
91:
79:logical matrix
48:linear algebra
34:The transpose
15:
9:
6:
4:
3:
2:
5333:
5322:
5319:
5317:
5314:
5312:
5309:
5308:
5306:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5225:
5223:
5219:
5209:
5206:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5165:
5163:
5159:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5127:Metric tensor
5125:
5123:
5120:
5118:
5115:
5114:
5112:
5108:
5105:
5101:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5042:
5040:
5039:Exterior form
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5010:
5007:
5005:
5002:
5001:
4999:
4993:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4932:
4930:
4928:
4924:
4918:
4915:
4913:
4912:Tensor bundle
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4869:
4867:
4861:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4811:
4809:
4805:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4771:
4769:
4764:
4761:
4759:
4756:
4755:
4752:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4707:
4705:
4703:
4699:
4696:
4692:
4688:
4687:
4681:
4677:
4670:
4665:
4663:
4658:
4656:
4651:
4650:
4647:
4635:
4627:
4626:
4623:
4617:
4614:
4612:
4611:Sparse matrix
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4593:
4591:
4589:
4585:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4550:
4548:
4546:constructions
4545:
4541:
4535:
4534:Outermorphism
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4494:Cross product
4492:
4490:
4487:
4486:
4484:
4482:
4478:
4472:
4469:
4467:
4464:
4462:
4461:Outer product
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4441:Orthogonality
4439:
4438:
4436:
4434:
4430:
4424:
4421:
4419:
4418:Cramer's rule
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4388:Decomposition
4386:
4384:
4381:
4380:
4378:
4376:
4372:
4367:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4263:
4261:
4257:
4251:
4248:
4246:
4243:
4242:
4239:
4235:
4228:
4223:
4221:
4216:
4214:
4209:
4208:
4205:
4198:
4194:
4193:
4183:
4181:0-486-42000-0
4177:
4173:
4172:
4166:
4162:
4158:
4154:
4148:
4144:
4140:
4136:
4132:
4128:
4124:
4118:
4114:
4110:
4106:
4102:
4098:
4092:
4088:
4087:
4081:
4077:
4071:
4067:
4063:
4059:
4058:
4053:
4049:
4045:
4039:
4035:
4027:
4023:
4022:
4017:
4013:
4012:
4000:
3995:
3988:
3987:Bourbaki 1989
3983:
3976:
3971:
3964:
3959:
3953:
3952:0-03-010567-6
3949:
3946:
3942:
3938:
3933:
3919:
3915:
3908:
3893:
3892:
3887:
3881:
3866:
3862:
3856:
3848:
3842:
3839:. CRC Press.
3838:
3837:
3829:
3822:
3818:
3814:
3808:
3793:
3789:
3782:
3778:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3749:
3746:
3745:
3739:
3737:
3733:
3728:
3724:
3720:
3710:
3704:
3700:
3696:
3688:
3684:
3680:
3673:
3671:
3667:
3658:
3652:
3646:
3637:
3628:
3624:
3618:
3614:
3590:
3587:
3581:
3575:
3563:
3559:
3555:
3542:
3536:
3533:
3530:
3518:
3514:
3506:
3505:
3504:
3500:
3496:
3492:
3482:
3467:
3466:vector spaces
3463:
3457:
3453:
3449:
3442:
3440:
3434:
3430:
3423:
3419:
3415:
3412:
3411:nondegenerate
3397:
3387:
3385:
3380:
3376:
3372:
3362:
3358:
3354:
3350:
3346:
3342:
3335:
3331:
3327:
3323:
3319:
3315:
3309:
3305:
3301:
3295:
3288:
3284:
3280:
3276:
3272:
3268:
3262:
3258:
3254:
3250:
3244:
3240:
3236:
3229:
3228:Bilinear form
3219:
3217:
3212:
3199:
3194:
3188:
3182:
3177:
3166:
3160:
3149:
3140:
3136:
3125:
3116:
3112:
3107:
3102:
3098:
3094:
3075:
3066:
3062:
3057:
3055:
3050:
3046:
3042:
3036:
3032:
3025:
3011:
3007:
3001:
2997:
2989:
2985:
2981:
2973:
2969:
2965:
2959:
2958:
2957:
2952:
2940:
2939:
2932:
2928:
2922:
2918:
2914:
2908:
2904:
2900:
2896:, is the map
2895:
2891:
2887:
2882:
2878:
2874:
2854:
2846:
2841:
2834:
2824:
2822:
2816:
2814:
2808:
2798:
2796:
2792:
2788:
2784:
2781: ≠
2780:
2776:
2772:
2768:
2764:
2761: ×
2760:
2755:
2753:
2749:
2745:
2740:
2738:
2734:
2730:
2726:
2722:
2715:
2710:
2705:
2681:
2661:
2639:
2624:
2619:
2609:
2587:
2578:
2577:
2576:
2573:
2567:
2564:
2561:
2545:
2538:
2525:
2504:
2498:
2492:
2486:
2480:
2475:
2474:inner product
2470:
2465:
2459:
2455:
2448:
2445:
2438:
2434:
2427:
2422:
2417:
2410:
2406:
2399:
2308:
2282:
2267:
2259:
2243:
2223:
2203:
2139:
2131:
2115:
2095:
2075:
2072:
2069:
2017:
2016:
2014:
2011:
1956:
1955:
1924:
1886:
1878:
1875:
1871:
1866:
1861:
1855:
1851:The notation
1848:
1847:
1834:
1824:
1819:
1816:
1806:
1801:
1796:
1793:
1788:
1773:
1764:
1761:
1756:
1753:
1747:
1742:
1728:
1708:
1700:
1686:
1680:
1675:
1671:
1666:
1662:
1661:
1648:
1631:
1627:
1612:
1601:
1590:
1584:
1578:
1574:
1570:
1566:
1561:
1557:
1552:
1548:
1541:
1535:
1530:
1529:
1527:
1526:
1513:
1486:
1476:
1465:
1456:
1449:
1445:
1437:
1433:
1427:
1423:
1419:
1418:
1405:
1390:
1387:
1377:
1368:
1364:
1355:
1350:
1346:
1345:
1332:
1317:
1302:
1292:
1283:
1274:
1265:
1260:
1256:
1252:
1251:
1238:
1230:
1220:
1205:
1196:
1195:
1194:
1192:
1183:
1177:
1150:
1144:
1139:
1134:
1127:
1122:
1117:
1111:
1106:
1095:
1089:
1084:
1077:
1072:
1065:
1060:
1054:
1044:
1043:
1026:
1020:
1015:
1008:
1003:
997:
992:
981:
975:
970:
963:
958:
952:
942:
941:
924:
918:
911:
905:
899:
888:
882:
877:
871:
861:
860:
840:
830:
827:
815:
796:
795:
794:
791:
786:
785:
765:
760:
757:
747:
728:
727:
726:
723:
718:
717:
713:is called an
712:
693:
678:
675:
656:
655:
654:
651:
646:
645:
640:
621:
606:
587:
586:
585:
582:
577:
573:
572:
567:
563:
544:
536:
533:
514:
513:
512:
509:
504:
503:
483:
475:
456:
455:
454:
451:
446:
445:
434:
432:
423:
420:
413:
409:
402:
396:
386:
380:
376:
372:
365:
361:matrix, then
358:
354:
347:
327:
322:
319:
314:
306:
301:
296:
293:
288:
273:
264:
263:
262:
259:
245:
228:
222:
217:
214:
208:
203:
200:
195:
194:main diagonal
190:
186:
183:
182:
181:
178:
172:
166:
160:
141:
137:
127:
121:
115:
111:, denoted by
109:
90:
88:
84:
80:
76:
75:Arthur Cayley
71:
68:
62:
57:
53:
49:
41:
37:
32:
26:
22:
5290:Hermann Weyl
5094:Vector space
5079:Pseudotensor
5044:Fiber bundle
4997:abstractions
4984:
4892:Mixed tensor
4877:Tensor field
4684:
4544:Vector space
4350:
4276:Vector space
4170:
4142:
4108:
4085:
4068:, Springer,
4065:
4062:Halmos, Paul
4025:
4020:
3994:
3982:
3970:
3958:
3940:
3932:
3921:. Retrieved
3917:
3907:
3895:. Retrieved
3889:
3880:
3868:. Retrieved
3864:
3855:
3835:
3828:
3820:
3816:
3807:
3797:September 8,
3795:. Retrieved
3792:Math Insight
3791:
3781:
3729:
3722:
3718:
3702:
3698:
3694:
3691:is equal to
3686:
3682:
3678:
3674:
3656:
3650:
3644:
3633:
3626:
3622:
3616:
3612:
3498:
3494:
3490:
3480:
3475:, we define
3455:
3451:
3447:
3443:
3438:
3432:
3428:
3421:
3417:
3400:
3378:
3374:
3371:homomorphism
3360:
3356:
3352:
3348:
3344:
3340:
3333:
3329:
3325:
3321:
3317:
3313:
3307:
3303:
3299:
3293:
3286:
3282:
3278:
3274:
3270:
3266:
3260:
3256:
3252:
3248:
3242:
3238:
3234:
3231:
3210:
3192:
3189:
3180:
3175:
3164:
3158:
3147:
3138:
3134:
3123:
3114:
3110:
3105:
3100:
3096:
3092:
3073:
3058:
3051:
3044:
3040:
3034:
3030:
3015:
3009:
3005:
2999:
2995:
2987:
2983:
2979:
2971:
2967:
2963:
2936:
2930:
2926:
2920:
2916:
2912:
2906:
2902:
2898:
2893:
2889:
2880:
2876:
2872:
2839:
2836:
2817:
2810:
2782:
2778:
2774:
2762:
2758:
2756:
2741:
2718:
2571:
2568:
2562:
2559:
2543:
2536:
2527:. The entry
2523:
2502:
2496:
2490:
2484:
2478:
2476:of a row of
2468:
2457:
2453:
2446:
2443:
2436:
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2425:
2415:
2408:
2404:
2397:
2394:
1864:
1853:
1754:
1751:
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1181:
1175:
1172:
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782:
780:
721:
714:
708:
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642:
636:
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578:); that is,
569:
559:
507:
500:
498:
449:
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440:
424:
418:
411:
407:
400:
394:
384:
381:
374:
370:
363:
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188:
176:
170:
164:
158:
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119:
113:
107:
104:
72:
66:
60:
51:
45:
39:
38:of a matrix
35:
5230:Élie Cartan
5178:Spin tensor
5152:Weyl tensor
5110:Mathematics
5074:Multivector
4865:definitions
4763:Engineering
4702:Mathematics
4524:Multivector
4489:Determinant
4446:Dot product
4291:Linear span
3999:Trèves 2006
3975:Halmos 1974
3945:Brooks/Cole
3914:"Transpose"
3670:orthonormal
3636:isomorphism
3384:double dual
3173:. The map
2910:defined by
2888:, then its
2843:denote the
2787:permutation
2413:matrix and
1870:eigenvalues
1674:dot product
1665:determinant
787:; that is,
719:; that is,
647:; that is,
505:; that is,
447:; that is,
416:, notation
5305:Categories
5059:Linear map
4927:Operations
4558:Direct sum
4393:Invertible
4296:Linear map
3923:2020-09-08
3773:References
3503:satisfies
3462:linear map
3216:dual bases
3122:) ⊆
2886:linear map
2805:See also:
2735:, such as
2702:See also:
1422:linear map
1255:involution
1169:Properties
101:Definition
5198:EM tensor
5034:Dimension
4985:Transpose
4588:Numerical
4351:Transpose
4161:853623322
4141:(2006) .
4131:840278135
4018:(1989) .
3989:, II §2.5
3865:ProofWiki
3382:into the
3365:. Here,
3181:transpose
3039: :=
2990:)⟩
2073:×
1817:−
1794:−
1701:⋅
1452:matrices.
1424:from the
836:¯
828:−
758:−
689:¯
679:−
637:A square
617:¯
560:A square
537:−
379:matrix.
251:-th row,
237:-th row,
192:over its
142:⊺
52:transpose
5311:Matrices
5064:Manifold
5049:Geodesic
4807:Notation
4634:Category
4573:Subspace
4568:Quotient
4519:Bivector
4433:Bilinear
4375:Matrices
4250:Glossary
4064:(1974),
4052:18588156
3742:See also
3697: :
3681: :
3638:between
3625:∈
3615:∈
3610:for all
3493: :
3464:between
3450: :
3302: :
3251: :
3237: :
3146:→
3137: :
3095: :
3037:⟩
3029:⟨
2993:for all
2978:⟨
2974:⟩
2962:⟨
2951:relation
2938:pullback
2905:→
2901: :
2875: :
2823:choice.
2767:in-place
2721:computer
2391:Products
2108:and let
1349:addition
857:Examples
431:variable
5161:Physics
4995:Related
4758:Physics
4676:Tensors
4245:Outline
3939:(2006)
3481:adjoint
3479:as the
3439:adjoint
3390:Adjoint
3022:is the
2858:. Let
2769:, with
2765:matrix
2256:is the
1923:similar
1259:inverse
711:inverse
639:complex
562:complex
247:is the
185:Reflect
50:, the
5089:Vector
5084:Spinor
5069:Matrix
4863:Tensor
4529:Tensor
4341:Kernel
4271:Vector
4266:Scalar
4178:
4159:
4149:
4129:
4119:
4093:
4072:
4050:
4040:
4028:]
3950:
3843:
3082:. If
3067:(TVS)
3016:where
2853:module
2847:of an
2777:. For
2725:memory
2401:is an
1257:(self-
1191:scalar
433:name.
367:is an
349:is an
56:matrix
5009:Basis
4694:Scope
4398:Minor
4383:Block
4321:Basis
4030:(PDF)
4024:[
3977:, §44
3897:4 Feb
3870:4 Feb
3666:bases
3460:is a
3311:i.e.
3198:bases
3169:'
3152:'
3143:'
3128:'
3119:'
3078:'
3063:of a
3054:below
2884:is a
2821:basis
2719:On a
2506:. If
2152:. If
2128:be a
1758:is a
1426:space
1189:be a
429:as a
410:) = (
54:of a
4553:Dual
4408:Rank
4176:ISBN
4157:OCLC
4147:ISBN
4127:OCLC
4117:ISBN
4091:ISBN
4070:ISBN
4048:OCLC
4038:ISBN
3948:ISBN
3899:2021
3872:2021
3841:ISBN
3799:2020
3668:are
3654:and
3642:and
3620:and
3471:and
3426:and
3405:and
3351:) =
3324:) =
3277:) =
3204:and
3086:and
3059:The
3003:and
2894:dual
2862:and
2837:Let
2771:O(1)
2737:BLAS
2731:for
2517:and
2441:and
2174:and
2040:and
2018:Let
1979:and
1682:and
1672:The
1663:The
1179:and
1173:Let
4113:GTM
3821:148
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3328:(Ψ(
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3008:∈
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2682:.
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2662:=
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2645:T
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2620:=
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2552:p
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743:T
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685:A
676:=
671:T
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622:.
613:A
607:=
602:T
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541:A
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476:=
471:T
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