4377:
2720:
4641:
2703:
2592:
2829:
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
42:
1174:
1354:
1535:
1050:
1856:
948:
2698:{\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} }.}
1427:
1260:
1750:
1670:
862:
1058:
715:
643:
3619:
1279:
787:
566:
505:
1470:
956:
2753:
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
2369:
2017:
1963:
1778:
875:
349:
1702:
165:
2391:
2340:
2314:
2205:
2183:
2071:
2049:
1988:
1930:
1369:
2097:
1210:
3749:
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.
53:
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.
2757:, 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
2289:
2265:
2245:
2225:
2161:
2137:
2117:
1908:
1615:
810:
1169:{\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}}
670:
601:
2750:, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
1349:{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.}
3683:
with respect to their bilinear forms. In this context, many authors however, use the term transpose to refer to the adjoint as defined here.
3520:
742:
528:
470:
3896:
5034:
4235:
1530:{\displaystyle \left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.}
1045:{\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3\\2&4\end{bmatrix}}}
1860:
The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.
4568:
4626:
1880:
5183:
4161:
4131:
4105:
4084:
4052:
3855:
3726:
with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps
108:
This article assumes that matrices are taken over a commutative ring. These results may not hold in the non-commutative case.
2768:
Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an
2023:, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties.
1431:
The transpose of a scalar is the same scalar. Together with the preceding property, this implies that the transpose is a
1851:{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\operatorname {T} }.}
5218:
4897:
3063:
The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint (
2345:
1993:
1939:
69:
is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix
4677:
4190:
3962:
2826:, the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps.
5099:
4616:
2948:
2761:
algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing
17:
4044:
1539:
The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so
4578:
4514:
3772:
2823:
943:{\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\operatorname {T} }=\,{\begin{bmatrix}1\\2\end{bmatrix}}}
409:. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as
35:
4950:
4882:
4123:
2801:
2714:
4207:
278:
4975:
4356:
4228:
1770:
5213:
4461:
4311:
3777:
2754:
2724:
5024:
4844:
4366:
4260:
5326:
4696:
4606:
4255:
2843:
2817:
1884:
31:
5178:
4030:
143:
5280:
5198:
5152:
4859:
4598:
4481:
3075:
1422:{\displaystyle \left(c\mathbf {A} \right)^{\operatorname {T} }=c\mathbf {A} ^{\operatorname {T} }.}
1265:
415:. An advantage of this notation is that no parentheses are needed when exponents are involved: as
195:
2374:
2323:
2297:
2188:
2166:
2054:
2032:
1971:
1913:
5331:
5250:
4937:
4854:
4824:
4644:
4573:
4351:
4221:
3845:
1255:{\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .}
5208:
5064:
5019:
4408:
4341:
4331:
3676:
3208:
2831:
2758:
2076:
441:
4180:
4095:
5290:
5245:
4725:
4670:
4423:
4418:
4413:
4346:
4291:
3823:
3071:
2431:
1745:{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} .}
654:
512:
5265:
5193:
5079:
4945:
4907:
4839:
4433:
4398:
4385:
4276:
4149:
2863:
2855:
2735:
1201:
66:
8:
5142:
4965:
4955:
4804:
4789:
4745:
4611:
4491:
4466:
4316:
3767:
586:
5275:
5132:
4985:
4799:
4735:
4321:
4115:
3421:
2739:
2474:
2317:
2274:
2250:
2230:
2210:
2146:
2122:
2102:
1893:
1665:{\displaystyle \det \left(\mathbf {A} ^{\operatorname {T} }\right)=\det(\mathbf {A} ).}
792:
A square complex matrix whose transpose is equal to its conjugate inverse is called a
5321:
5270:
5039:
5014:
4829:
4740:
4720:
4519:
4476:
4403:
4296:
4186:
4167:
4157:
4137:
4127:
4101:
4080:
4058:
4048:
3958:
3851:
3746:
3742:
3406:
2800:
of the data elements that is non-trivial to implement in-place. Therefore, efficient
2268:
2020:
1933:
726:
652:
matrix whose transpose is equal to the negation of its complex conjugate is called a
576:
97:
3871:
5285:
4960:
4927:
4912:
4794:
4663:
4524:
4428:
4281:
4126:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
4026:
3719:
2805:
581:
454:
857:{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} ^{-1}}}.}
5255:
5203:
5147:
5127:
5029:
4917:
4784:
4755:
4583:
4376:
4336:
4326:
3762:
3758:
3060:). This definition also applies unchanged to left modules and to vector spaces.
3034:
2961:
2762:
2140:
1359:
93:
5295:
5260:
5157:
4990:
4980:
4970:
4892:
4864:
4849:
4834:
4750:
4588:
4509:
4244:
3947:
3924:
3901:
3680:
2781:
2743:
1269:
794:
721:
649:
575:
matrix whose transpose is equal to the matrix with every entry replaced by its
572:
89:
58:
5240:
2830:
the transpose of the matrix representing the linear map, independently of the
710:{\displaystyle \mathbf {A} ^{\operatorname {T} }=-{\overline {\mathbf {A} }}.}
84:
The transpose of a matrix was introduced in 1858 by the
British mathematician
5315:
5232:
5137:
5049:
4922:
4621:
4544:
4504:
4471:
4451:
4171:
4141:
3424:
3238:
2719:
2484:
638:{\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} }}.}
204:
85:
4062:
5300:
5104:
5089:
5054:
4902:
4887:
4554:
4443:
4393:
4286:
3798:
3675:. The matrix of the adjoint of a map is the transposed matrix only if the
3476:
3381:
1436:
2808:, starting in the late 1950s, and several algorithms have been developed.
5188:
5162:
5084:
4773:
4712:
4534:
4499:
4456:
4301:
4072:
3955:
3834: : 17–37. The transpose (or "transposition") is defined on page 31.
3646:
3394:
2797:
1883:
are equal to the eigenvalues of its transpose, since they share the same
1684:
1675:
3302:. By defining the transpose of this bilinear form as the bilinear form
5069:
4563:
4306:
3614:{\displaystyle B_{X}{\big (}x,g(y){\big )}=B_{Y}{\big (}u(x),y{\big )}}
3472:
3226:
2896:
1432:
2708:
1868:
is sometimes used to represent either of these equivalent expressions.
5044:
2738:
by simply accessing the same data in a different order. For example,
510:
A square matrix whose transpose is equal to its negative is called a
782:{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} ^{-1}.}
436:
In this article this confusion is avoided by never using the symbol
5074:
5059:
4529:
3745:(conjugate-linear in one argument) instead of bilinear forms. The
3671:, resulting in an isomorphism between the transpose and adjoint of
2777:
2731:
1678:
of a square matrix is the same as the determinant of its transpose.
41:
4768:
4730:
4213:
2822:
As the main use of matrices is to represent linear maps between
561:{\displaystyle \mathbf {A} ^{\operatorname {T} }=-\mathbf {A} .}
5094:
4686:
4539:
3225:
describes the transpose of that linear map with respect to the
2811:
2511:, so the entry corresponds to the inner product of two rows of
500:{\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} .}
452:
A square matrix whose transpose is equal to itself is called a
2393:
is a single Jordan block, which is a straightforward exercise.
2026:
A proof of this property uses the following two observations.
3452:, which is closely related to the transpose, may be defined:
447:
3967:
2747:
4655:
4047:. Berlin New York: Springer Science & Business Media.
2804:
has been the subject of numerous research publications in
191:, may be constructed by any one of the following methods:
3828:
2734:, one can often avoid explicitly transposing a matrix in
1699:
can be computed as the single entry of the matrix product
27:
Matrix operation which flips a matrix over its diagonal
3897:"What is the best symbol for vector/matrix transpose?"
2524:
is the entry of the product, it is obtained from rows
1125:
1068:
1011:
966:
919:
885:
4003:
3523:
2784:
additional storage or at most storage much less than
2595:
2377:
2371:. This further reduces to proving the same fact when
2348:
2326:
2300:
2277:
2253:
2233:
2213:
2191:
2169:
2149:
2125:
2105:
2079:
2057:
2035:
1996:
1974:
1942:
1916:
1896:
1781:
1705:
1618:
1473:
1372:
1282:
1213:
1061:
959:
878:
813:
745:
673:
604:
531:
473:
281:
207:(which runs from top-left to bottom-right) to obtain
146:
4154:
Topological Vector Spaces, Distributions and
Kernels
2586:
results from the fact that it is its own transpose:
30:
This article is about the transpose of matrices and
2709:
Implementation of matrix transposition on computers
2434:with these two matrices gives two square matrices:
3741:Over a complex vector space, one often works with
3613:
2697:
2385:
2363:
2334:
2316:is a matrix over an algebraically closed field in
2308:
2283:
2259:
2239:
2219:
2199:
2177:
2155:
2131:
2111:
2091:
2065:
2043:
2011:
1982:
1957:
1924:
1902:
1850:
1744:
1664:
1529:
1421:
1348:
1254:
1168:
1044:
942:
856:
781:
709:
637:
560:
499:
343:
159:
2364:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
2012:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
1958:{\displaystyle \mathbf {A} ^{\operatorname {T} }}
5313:
1645:
1619:
720:A square matrix whose transpose is equal to its
3232:
75:by producing another matrix, often denoted by
4671:
4229:
4114:
3973:
3843:
3606:
3581:
3561:
3536:
585:(equivalent to the matrix being equal to its
2837:
2812:Transposes of linear maps and bilinear forms
1264:The operation of taking the transpose is an
579:(denoted here with an overline) is called a
4100:. San José: Solar Crest. pp. 122–132.
3872:"Transpose of a Matrix Product (ProofWiki)"
3847:Introduction to Linear Algebra, 2nd edition
3837:
4678:
4664:
4236:
4222:
3738:for which the adjoint equals the inverse.
3686:The adjoint allows us to consider whether
448:Matrix definitions involving transposition
913:
5035:Covariance and contravariance of vectors
4178:
4093:
4025:
3997:
2718:
103:
40:
3207:describes a linear map with respect to
2964:characterizes the algebraic adjoint of
2568:) is symmetric. Similarly, the product
2545:is also obtained from these rows, thus
14:
5314:
4627:Comparison of linear algebra libraries
4148:
4071:
4009:
3985:
344:{\displaystyle \left_{ij}=\left_{ji}.}
4659:
4217:
4156:. Mineola, N.Y.: Dover Publications.
3922:
2430:is its transpose, then the result of
4185:. Mineola: Dover. pp. 126–132.
4182:Introduction to Matrices and Vectors
3824:"A memoir on the theory of matrices"
96:R, the transpose corresponds to the
3952:Linear Algebra and its Applications
3243:Every linear map to the dual space
24:
4898:Tensors in curvilinear coordinates
4243:
4019:
3796:
3718:. In particular, this allows the
2687:
2667:
2655:
2645:
2625:
2614:
2473:. Furthermore, these products are
2356:
2247:. In particular this applies when
2004:
1950:
1840:
1794:
1729:
1633:
1519:
1507:
1492:
1411:
1393:
1338:
1323:
1308:
1236:
1226:
1112:
998:
905:
821:
753:
681:
612:
539:
481:
294:
25:
5343:
4200:
2580:A quick proof of the symmetry of
2320:with respect to some basis, then
4640:
4639:
4617:Basic Linear Algebra Subprograms
4375:
4077:Finite dimensional vector spaces
3030:⟨•, •⟩
2824:finite-dimensional vector spaces
2682:
2676:
2662:
2640:
2609:
2603:
2379:
2351:
2328:
2302:
2193:
2171:
2059:
2037:
1999:
1976:
1945:
1918:
1822:
1789:
1735:
1724:
1715:
1707:
1652:
1628:
1514:
1502:
1483:
1480:
1406:
1383:
1333:
1318:
1298:
1290:
1245:
1221:
833:
816:
763:
748:
695:
676:
623:
607:
551:
534:
490:
476:
393:In the case of square matrices,
321:
289:
4515:Seven-dimensional cross product
3991:
3645:These bilinear forms define an
3979:
3954:4th edition, page 51, Thomson
3941:
3916:
3889:
3864:
3844:T.A. Whitelaw (1 April 1991).
3816:
3790:
3595:
3589:
3556:
3550:
2796:, this involves a complicated
2099:matrices over some base field
1656:
1648:
160:{\displaystyle A^{\intercal }}
36:Transposition (disambiguation)
13:
1:
4951:Exterior covariant derivative
4883:Tensor (intrinsic definition)
4206:Gilbert Strang (Spring 2010)
3783:
2802:in-place matrix transposition
2715:In-place matrix transposition
2477:. Indeed, the matrix product
2227:, then they are similar over
2207:are similar as matrices over
1879:is a square matrix, then its
1451:matrices to the space of the
1179:
111:
4976:Raising and lowering indices
4357:Eigenvalues and eigenvectors
4118:; Wolff, Manfred P. (1999).
3233:Transpose of a bilinear form
3064:
2935:. The resulting functional
2386:{\displaystyle \mathbf {A} }
2335:{\displaystyle \mathbf {A} }
2309:{\displaystyle \mathbf {A} }
2200:{\displaystyle \mathbf {B} }
2178:{\displaystyle \mathbf {A} }
2066:{\displaystyle \mathbf {B} }
2044:{\displaystyle \mathbf {A} }
1983:{\displaystyle \mathbf {A} }
1925:{\displaystyle \mathbf {A} }
1771:positive-semidefinite matrix
1760:has only real entries, then
846:
699:
627:
7:
5214:Gluon field strength tensor
4685:
4179:Schwartz, Jacob T. (2001).
4094:Maruskin, Jared M. (2012).
3799:"The transpose of a matrix"
3778:Projection (linear algebra)
3773:Moore–Penrose pseudoinverse
3752:
3101:are TVSs then a linear map
2725:row- and column-major order
2401:
867:
10:
5348:
5025:Cartan formalism (physics)
4845:Penrose graphical notation
3404:
3400:
3236:
3167:denote the restriction of
2841:
2815:
2712:
2561:, and the product matrix (
116:The transpose of a matrix
107:
81:(among other notations).
29:
5231:
5171:
5120:
5113:
5005:
4936:
4873:
4817:
4764:
4711:
4704:
4697:Glossary of tensor theory
4693:
4635:
4597:
4553:
4490:
4442:
4384:
4373:
4269:
4251:
4120:Topological Vector Spaces
3974:Schaefer & Wolff 1999
3448:, a concept known as the
3308:defined by the transpose
2844:Transpose of a linear map
2838:Transpose of a linear map
2818:Transpose of a linear map
2483:has entries that are the
2092:{\displaystyle n\times n}
1885:characteristic polynomial
5281:Gregorio Ricci-Curbastro
5153:Riemann curvature tensor
4860:Van der Waerden notation
4210:from MIT Open Courseware
4097:Essential Linear Algebra
4045:Éléments de mathématique
4037:Algèbre: Chapitres 1 à 3
3405:Not to be confused with
3257:defines a bilinear form
3076:topological vector space
5251:Elwin Bruno Christoffel
5184:Angular momentum tensor
4855:Tetrad (index notation)
4825:Abstract index notation
3761:, the transpose of the
3143:, in which case we let
2577:is a symmetric matrix.
1358:The transpose respects
403:th power of the matrix
5065:Levi-Civita connection
4342:Row and column vectors
4032:Algebra I Chapters 1-3
3615:
2759:fast Fourier transform
2727:
2699:
2387:
2365:
2336:
2310:
2285:
2261:
2241:
2221:
2201:
2179:
2157:
2133:
2113:
2093:
2067:
2045:
2013:
1984:
1959:
1926:
1904:
1852:
1746:
1687:of two column vectors
1666:
1531:
1423:
1350:
1256:
1170:
1046:
944:
858:
783:
711:
639:
562:
501:
345:
266:-th column element of
252:-th column element of
161:
54:
34:. For other uses, see
5291:Jan Arnoldus Schouten
5246:Augustin-Louis Cauchy
4726:Differential geometry
4347:Row and column spaces
4292:Scalar multiplication
3929:mathworld.wolfram.com
3822:Arthur Cayley (1858)
3616:
3412:If the vector spaces
3072:continuous dual space
2722:
2700:
2499:. But the columns of
2432:matrix multiplication
2388:
2366:
2337:
2311:
2286:
2262:
2242:
2222:
2202:
2180:
2158:
2134:
2114:
2094:
2068:
2046:
2014:
1985:
1960:
1927:
1905:
1853:
1747:
1667:
1532:
1424:
1351:
1257:
1171:
1047:
945:
859:
784:
712:
664:is skew-Hermitian if
655:skew-Hermitian matrix
640:
563:
522:is skew-symmetric if
513:skew-symmetric matrix
502:
346:
229:Write the columns of
162:
104:Transpose of a matrix
44:
5266:Carl Friedrich Gauss
5199:stress–energy tensor
5194:Cauchy stress tensor
4946:Covariant derivative
4908:Antisymmetric tensor
4840:Multi-index notation
4482:Gram–Schmidt process
4434:Gaussian elimination
3722:over a vector space
3521:
3275:, with the relation
2856:algebraic dual space
2593:
2375:
2346:
2324:
2298:
2275:
2251:
2231:
2211:
2189:
2167:
2147:
2123:
2103:
2077:
2055:
2033:
1994:
1972:
1940:
1914:
1894:
1779:
1703:
1616:
1471:
1370:
1280:
1211:
1059:
957:
876:
811:
743:
671:
602:
529:
471:
399:may also denote the
279:
144:
5143:Nonmetricity tensor
4998:(2nd-order tensors)
4966:Hodge star operator
4956:Exterior derivative
4805:Transport phenomena
4790:Continuum mechanics
4746:Multilinear algebra
4612:Numerical stability
4492:Multilinear algebra
4467:Inner product space
4317:Linear independence
4116:Schaefer, Helmut H.
3923:Weisstein, Eric W.
3768:Conjugate transpose
587:conjugate transpose
88:. In the case of a
5276:Tullio Levi-Civita
5219:Metric tensor (GR)
5133:Levi-Civita symbol
4986:Tensor contraction
4800:General relativity
4736:Euclidean geometry
4322:Linear combination
3743:sesquilinear forms
3611:
3420:have respectively
3219:, then the matrix
2740:software libraries
2728:
2695:
2475:symmetric matrices
2383:
2361:
2332:
2318:Jordan normal form
2306:
2281:
2257:
2237:
2217:
2197:
2175:
2153:
2129:
2109:
2089:
2063:
2041:
2009:
1980:
1968:This implies that
1955:
1922:
1910:, a square matrix
1900:
1848:
1742:
1662:
1527:
1419:
1346:
1252:
1166:
1160:
1105:
1042:
1036:
991:
940:
934:
898:
854:
779:
707:
635:
558:
497:
433:is not ambiguous.
341:
221:as the columns of
215:Write the rows of
157:
55:
5309:
5308:
5271:Hermann Grassmann
5227:
5226:
5179:Moment of inertia
5040:Differential form
5015:Affine connection
4830:Einstein notation
4813:
4812:
4741:Exterior calculus
4721:Coordinate system
4653:
4652:
4520:Geometric algebra
4477:Kronecker product
4312:Linear projection
4297:Vector projection
4163:978-0-486-45352-1
4133:978-1-4612-7155-0
4107:978-0-9850627-3-6
4086:978-0-387-90093-3
4054:978-3-540-64243-5
4027:Bourbaki, Nicolas
3857:978-0-7514-0159-2
3747:Hermitian adjoint
3407:Hermitian adjoint
3117:weakly continuous
3037:(i.e. defined by
2960:. The following
2901:algebraic adjoint
2493:with a column of
2284:{\displaystyle k}
2269:algebraic closure
2260:{\displaystyle L}
2240:{\displaystyle k}
2220:{\displaystyle L}
2156:{\displaystyle k}
2132:{\displaystyle L}
2112:{\displaystyle k}
2021:invariant factors
1903:{\displaystyle k}
849:
736:is orthogonal if
727:orthogonal matrix
702:
630:
577:complex conjugate
98:converse relation
16:(Redirected from
5339:
5327:Abstract algebra
5286:Bernhard Riemann
5118:
5117:
4961:Exterior product
4928:Two-point tensor
4913:Symmetric tensor
4795:Electromagnetism
4709:
4708:
4680:
4673:
4666:
4657:
4656:
4643:
4642:
4525:Exterior algebra
4462:Hadamard product
4379:
4367:Linear equations
4238:
4231:
4224:
4215:
4214:
4196:
4175:
4150:Trèves, François
4145:
4111:
4089:
4066:
4042:
4013:
4007:
4001:
3995:
3989:
3983:
3977:
3971:
3965:
3945:
3939:
3938:
3936:
3935:
3920:
3914:
3913:
3911:
3909:
3893:
3887:
3886:
3884:
3882:
3868:
3862:
3861:
3841:
3835:
3820:
3814:
3813:
3811:
3809:
3794:
3737:
3736:
3725:
3720:orthogonal group
3717:
3716:
3701:
3700:
3674:
3670:
3664:
3658:
3652:
3640:
3630:
3620:
3618:
3617:
3612:
3610:
3609:
3585:
3584:
3578:
3577:
3565:
3564:
3540:
3539:
3533:
3532:
3513:
3512:
3497:
3489:
3485:
3481:
3470:
3469:
3447:
3436:
3419:
3415:
3392:
3379:
3375:
3348:
3321:
3307:
3301:
3274:
3256:
3224:
3218:
3214:
3206:
3197:
3189:
3183:
3182:
3181:
3172:
3166:
3165:
3164:
3156:
3155:
3142:
3141:
3140:
3132:
3131:
3114:
3100:
3096:
3092:
3091:
3090:
3081:
3059:
3049:
3032:
3031:
3023:
3013:
3003:
3002:
2986:
2967:
2959:
2955:
2945:
2934:
2920:
2894:
2880:
2876:
2872:
2868:
2861:
2853:
2806:computer science
2723:Illustration of
2704:
2702:
2701:
2696:
2691:
2690:
2685:
2679:
2671:
2670:
2665:
2659:
2658:
2653:
2649:
2648:
2643:
2629:
2628:
2623:
2619:
2618:
2617:
2612:
2606:
2585:
2576:
2567:
2560:
2544:
2537:
2531:
2527:
2523:
2516:
2510:
2505:are the rows of
2504:
2498:
2492:
2482:
2472:
2471:
2460:
2451:
2450:
2439:
2429:
2423:
2422:
2411:
2392:
2390:
2389:
2384:
2382:
2370:
2368:
2367:
2362:
2360:
2359:
2354:
2341:
2339:
2338:
2333:
2331:
2315:
2313:
2312:
2307:
2305:
2290:
2288:
2287:
2282:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2226:
2224:
2223:
2218:
2206:
2204:
2203:
2198:
2196:
2184:
2182:
2181:
2176:
2174:
2162:
2160:
2159:
2154:
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2072:
2070:
2069:
2064:
2062:
2050:
2048:
2047:
2042:
2040:
2018:
2016:
2015:
2010:
2008:
2007:
2002:
1989:
1987:
1986:
1981:
1979:
1964:
1962:
1961:
1956:
1954:
1953:
1948:
1931:
1929:
1928:
1923:
1921:
1909:
1907:
1906:
1901:
1878:
1867:
1857:
1855:
1854:
1849:
1844:
1843:
1838:
1834:
1833:
1825:
1811:
1810:
1802:
1798:
1797:
1792:
1768:
1759:
1751:
1749:
1748:
1743:
1738:
1733:
1732:
1727:
1718:
1710:
1698:
1692:
1671:
1669:
1668:
1663:
1655:
1641:
1637:
1636:
1631:
1605:
1536:
1534:
1533:
1528:
1523:
1522:
1517:
1511:
1510:
1505:
1496:
1495:
1490:
1486:
1462:
1461:
1450:
1449:
1428:
1426:
1425:
1420:
1415:
1414:
1409:
1397:
1396:
1391:
1387:
1386:
1355:
1353:
1352:
1347:
1342:
1341:
1336:
1327:
1326:
1321:
1312:
1311:
1306:
1302:
1301:
1293:
1261:
1259:
1258:
1253:
1248:
1240:
1239:
1234:
1230:
1229:
1224:
1199:
1196:be matrices and
1195:
1189:
1175:
1173:
1172:
1167:
1165:
1164:
1116:
1115:
1110:
1109:
1051:
1049:
1048:
1043:
1041:
1040:
1002:
1001:
996:
995:
949:
947:
946:
941:
939:
938:
909:
908:
903:
902:
863:
861:
860:
855:
850:
845:
844:
836:
830:
825:
824:
819:
803:
788:
786:
785:
780:
775:
774:
766:
757:
756:
751:
735:
716:
714:
713:
708:
703:
698:
693:
685:
684:
679:
663:
644:
642:
641:
636:
631:
626:
621:
616:
615:
610:
595:is Hermitian if
594:
582:Hermitian matrix
567:
565:
564:
559:
554:
543:
542:
537:
521:
506:
504:
503:
498:
493:
485:
484:
479:
464:is symmetric if
463:
455:symmetric matrix
439:
432:
426:
414:
408:
402:
398:
389:
388:
377:
371:
370:
359:
350:
348:
347:
342:
337:
336:
328:
324:
311:
310:
302:
298:
297:
292:
271:
265:
261:
257:
251:
247:
240:
234:
226:
220:
212:
202:
190:
184:
178:
172:
166:
164:
163:
158:
156:
155:
139:
133:
127:
121:
80:
74:
32:linear operators
21:
18:Transpose matrix
5347:
5346:
5342:
5341:
5340:
5338:
5337:
5336:
5312:
5311:
5310:
5305:
5256:Albert Einstein
5223:
5204:Einstein tensor
5167:
5148:Ricci curvature
5128:Kronecker delta
5114:Notable tensors
5109:
5030:Connection form
5007:
5001:
4932:
4918:Tensor operator
4875:
4869:
4809:
4785:Computer vision
4778:
4760:
4756:Tensor calculus
4700:
4689:
4684:
4654:
4649:
4631:
4593:
4549:
4486:
4438:
4380:
4371:
4337:Change of basis
4327:Multilinear map
4265:
4247:
4242:
4203:
4193:
4164:
4134:
4108:
4087:
4055:
4040:
4022:
4020:Further reading
4017:
4016:
4008:
4004:
3996:
3992:
3984:
3980:
3972:
3968:
3946:
3942:
3933:
3931:
3921:
3917:
3907:
3905:
3895:
3894:
3890:
3880:
3878:
3870:
3869:
3865:
3858:
3842:
3838:
3821:
3817:
3807:
3805:
3797:Nykamp, Duane.
3795:
3791:
3786:
3763:cofactor matrix
3759:Adjugate matrix
3755:
3728:
3727:
3723:
3704:
3703:
3688:
3687:
3672:
3666:
3660:
3654:
3650:
3632:
3622:
3605:
3604:
3580:
3579:
3573:
3569:
3560:
3559:
3535:
3534:
3528:
3524:
3522:
3519:
3518:
3500:
3499:
3495:
3487:
3483:
3479:
3457:
3456:
3446:
3438:
3435:
3427:
3417:
3413:
3410:
3403:
3384:
3380:is the natural
3377:
3350:
3349:, we find that
3323:
3309:
3303:
3276:
3258:
3244:
3241:
3235:
3220:
3216:
3212:
3202:
3195:
3185:
3179:
3178:
3174:
3168:
3162:
3161:
3153:
3152:
3144:
3138:
3137:
3129:
3128:
3120:
3119:if and only if
3102:
3098:
3094:
3088:
3087:
3083:
3079:
3039:
3038:
3035:natural pairing
3029:
3028:
3015:
3005:
2988:
2972:
2971:
2965:
2957:
2953:
2936:
2922:
2908:
2882:
2878:
2874:
2870:
2866:
2859:
2849:
2846:
2840:
2820:
2814:
2763:memory locality
2755:row-major order
2717:
2711:
2686:
2681:
2680:
2675:
2666:
2661:
2660:
2654:
2644:
2639:
2638:
2634:
2633:
2624:
2613:
2608:
2607:
2602:
2601:
2597:
2596:
2594:
2591:
2590:
2581:
2569:
2566:
2562:
2559:
2552:
2546:
2543:
2539:
2533:
2529:
2525:
2522:
2518:
2512:
2506:
2500:
2494:
2488:
2478:
2463:
2462:
2453:
2442:
2441:
2435:
2425:
2414:
2413:
2407:
2404:
2378:
2376:
2373:
2372:
2355:
2350:
2349:
2347:
2344:
2343:
2327:
2325:
2322:
2321:
2301:
2299:
2296:
2295:
2276:
2273:
2272:
2252:
2249:
2248:
2232:
2229:
2228:
2212:
2209:
2208:
2192:
2190:
2187:
2186:
2170:
2168:
2165:
2164:
2148:
2145:
2144:
2141:field extension
2124:
2121:
2120:
2104:
2101:
2100:
2078:
2075:
2074:
2058:
2056:
2053:
2052:
2036:
2034:
2031:
2030:
2003:
1998:
1997:
1995:
1992:
1991:
1975:
1973:
1970:
1969:
1949:
1944:
1943:
1941:
1938:
1937:
1917:
1915:
1912:
1911:
1895:
1892:
1891:
1890:Over any field
1874:
1863:
1861:
1839:
1826:
1821:
1820:
1816:
1815:
1803:
1793:
1788:
1787:
1783:
1782:
1780:
1777:
1776:
1761:
1755:
1734:
1728:
1723:
1722:
1714:
1706:
1704:
1701:
1700:
1694:
1688:
1651:
1632:
1627:
1626:
1622:
1617:
1614:
1613:
1604:
1598:
1591:
1582:
1573:
1565:
1555:
1549:
1542:
1518:
1513:
1512:
1506:
1501:
1500:
1491:
1479:
1475:
1474:
1472:
1469:
1468:
1453:
1452:
1441:
1440:
1410:
1405:
1404:
1392:
1382:
1378:
1374:
1373:
1371:
1368:
1367:
1337:
1332:
1331:
1322:
1317:
1316:
1307:
1297:
1289:
1288:
1284:
1283:
1281:
1278:
1277:
1244:
1235:
1225:
1220:
1219:
1215:
1214:
1212:
1209:
1208:
1197:
1191:
1185:
1182:
1159:
1158:
1153:
1148:
1142:
1141:
1136:
1131:
1121:
1120:
1111:
1104:
1103:
1098:
1092:
1091:
1086:
1080:
1079:
1074:
1064:
1063:
1062:
1060:
1057:
1056:
1035:
1034:
1029:
1023:
1022:
1017:
1007:
1006:
997:
990:
989:
984:
978:
977:
972:
962:
961:
960:
958:
955:
954:
933:
932:
926:
925:
915:
914:
904:
897:
896:
891:
881:
880:
879:
877:
874:
873:
870:
837:
832:
831:
829:
820:
815:
814:
812:
809:
808:
799:
767:
762:
761:
752:
747:
746:
744:
741:
740:
731:
694:
692:
680:
675:
674:
672:
669:
668:
659:
622:
620:
611:
606:
605:
603:
600:
599:
590:
550:
538:
533:
532:
530:
527:
526:
517:
489:
480:
475:
474:
472:
469:
468:
459:
450:
437:
428:
416:
410:
404:
400:
394:
380:
379:
373:
362:
361:
355:
329:
320:
316:
315:
303:
293:
288:
287:
283:
282:
280:
277:
276:
267:
263:
259:
253:
249:
245:
236:
235:as the rows of
230:
222:
216:
208:
198:
186:
180:
174:
168:
151:
147:
145:
142:
141:
135:
129:
123:
117:
114:
109:
106:
94:binary relation
92:representing a
76:
70:
39:
28:
23:
22:
15:
12:
11:
5:
5345:
5335:
5334:
5332:Linear algebra
5329:
5324:
5307:
5306:
5304:
5303:
5298:
5296:Woldemar Voigt
5293:
5288:
5283:
5278:
5273:
5268:
5263:
5261:Leonhard Euler
5258:
5253:
5248:
5243:
5237:
5235:
5233:Mathematicians
5229:
5228:
5225:
5224:
5222:
5221:
5216:
5211:
5206:
5201:
5196:
5191:
5186:
5181:
5175:
5173:
5169:
5168:
5166:
5165:
5160:
5158:Torsion tensor
5155:
5150:
5145:
5140:
5135:
5130:
5124:
5122:
5115:
5111:
5110:
5108:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5022:
5017:
5011:
5009:
5003:
5002:
5000:
4999:
4993:
4991:Tensor product
4988:
4983:
4981:Symmetrization
4978:
4973:
4971:Lie derivative
4968:
4963:
4958:
4953:
4948:
4942:
4940:
4934:
4933:
4931:
4930:
4925:
4920:
4915:
4910:
4905:
4900:
4895:
4893:Tensor density
4890:
4885:
4879:
4877:
4871:
4870:
4868:
4867:
4865:Voigt notation
4862:
4857:
4852:
4850:Ricci calculus
4847:
4842:
4837:
4835:Index notation
4832:
4827:
4821:
4819:
4815:
4814:
4811:
4810:
4808:
4807:
4802:
4797:
4792:
4787:
4781:
4779:
4777:
4776:
4771:
4765:
4762:
4761:
4759:
4758:
4753:
4751:Tensor algebra
4748:
4743:
4738:
4733:
4731:Dyadic algebra
4728:
4723:
4717:
4715:
4706:
4702:
4701:
4694:
4691:
4690:
4683:
4682:
4675:
4668:
4660:
4651:
4650:
4648:
4647:
4636:
4633:
4632:
4630:
4629:
4624:
4619:
4614:
4609:
4607:Floating-point
4603:
4601:
4595:
4594:
4592:
4591:
4589:Tensor product
4586:
4581:
4576:
4574:Function space
4571:
4566:
4560:
4558:
4551:
4550:
4548:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4510:Triple product
4507:
4502:
4496:
4494:
4488:
4487:
4485:
4484:
4479:
4474:
4469:
4464:
4459:
4454:
4448:
4446:
4440:
4439:
4437:
4436:
4431:
4426:
4424:Transformation
4421:
4416:
4414:Multiplication
4411:
4406:
4401:
4396:
4390:
4388:
4382:
4381:
4374:
4372:
4370:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4273:
4271:
4270:Basic concepts
4267:
4266:
4264:
4263:
4258:
4252:
4249:
4248:
4245:Linear algebra
4241:
4240:
4233:
4226:
4218:
4212:
4211:
4208:Linear Algebra
4202:
4201:External links
4199:
4198:
4197:
4191:
4176:
4162:
4146:
4132:
4112:
4106:
4091:
4085:
4068:
4067:
4053:
4021:
4018:
4015:
4014:
4012:, p. 240.
4002:
3990:
3978:
3976:, p. 128.
3966:
3948:Gilbert Strang
3940:
3915:
3902:Stack Exchange
3888:
3863:
3856:
3836:
3815:
3788:
3787:
3785:
3782:
3781:
3780:
3775:
3770:
3765:
3754:
3751:
3659:, and between
3643:
3642:
3608:
3603:
3600:
3597:
3594:
3591:
3588:
3583:
3576:
3572:
3568:
3563:
3558:
3555:
3552:
3549:
3546:
3543:
3538:
3531:
3527:
3442:
3431:
3425:bilinear forms
3402:
3399:
3237:Main article:
3234:
3231:
3201:If the matrix
3190:is called the
3082:is denoted by
3025:
3024:
2946:is called the
2881:-modules. If
2842:Main article:
2839:
2836:
2813:
2810:
2744:linear algebra
2710:
2707:
2706:
2705:
2694:
2689:
2684:
2678:
2674:
2669:
2664:
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2616:
2611:
2605:
2600:
2564:
2557:
2550:
2541:
2520:
2403:
2400:
2399:
2398:
2397:
2396:
2395:
2394:
2381:
2358:
2353:
2342:is similar to
2330:
2304:
2292:
2280:
2256:
2236:
2216:
2195:
2173:
2152:
2128:
2108:
2088:
2085:
2082:
2061:
2039:
2024:
2019:have the same
2006:
2001:
1978:
1952:
1947:
1920:
1899:
1888:
1871:
1870:
1869:
1847:
1842:
1837:
1832:
1829:
1824:
1819:
1814:
1809:
1806:
1801:
1796:
1791:
1786:
1774:
1752:
1741:
1737:
1731:
1726:
1721:
1717:
1713:
1709:
1681:
1680:
1679:
1661:
1658:
1654:
1650:
1647:
1644:
1640:
1635:
1630:
1625:
1621:
1611:
1610:
1609:
1608:
1607:
1602:
1596:
1586:
1578:
1574:) =
1569:
1560:
1553:
1547:
1526:
1521:
1516:
1509:
1504:
1499:
1494:
1489:
1485:
1482:
1478:
1466:
1465:
1464:
1418:
1413:
1408:
1403:
1400:
1395:
1390:
1385:
1381:
1377:
1365:
1364:
1363:
1345:
1340:
1335:
1330:
1325:
1320:
1315:
1310:
1305:
1300:
1296:
1292:
1287:
1275:
1274:
1273:
1251:
1247:
1243:
1238:
1233:
1228:
1223:
1218:
1181:
1178:
1177:
1176:
1163:
1157:
1154:
1152:
1149:
1147:
1144:
1143:
1140:
1137:
1135:
1132:
1130:
1127:
1126:
1124:
1119:
1114:
1108:
1102:
1099:
1097:
1094:
1093:
1090:
1087:
1085:
1082:
1081:
1078:
1075:
1073:
1070:
1069:
1067:
1053:
1052:
1039:
1033:
1030:
1028:
1025:
1024:
1021:
1018:
1016:
1013:
1012:
1010:
1005:
1000:
994:
988:
985:
983:
980:
979:
976:
973:
971:
968:
967:
965:
951:
950:
937:
931:
928:
927:
924:
921:
920:
918:
912:
907:
901:
895:
892:
890:
887:
886:
884:
869:
866:
865:
864:
853:
848:
843:
840:
835:
828:
823:
818:
804:is unitary if
795:unitary matrix
790:
789:
778:
773:
770:
765:
760:
755:
750:
718:
717:
706:
701:
697:
691:
688:
683:
678:
646:
645:
634:
629:
625:
619:
614:
609:
569:
568:
557:
553:
549:
546:
541:
536:
508:
507:
496:
492:
488:
483:
478:
449:
446:
352:
351:
340:
335:
332:
327:
323:
319:
314:
309:
306:
301:
296:
291:
286:
244:Formally, the
242:
241:
227:
213:
154:
150:
113:
110:
105:
102:
90:logical matrix
59:linear algebra
45:The transpose
26:
9:
6:
4:
3:
2:
5344:
5333:
5330:
5328:
5325:
5323:
5320:
5319:
5317:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5238:
5236:
5234:
5230:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5200:
5197:
5195:
5192:
5190:
5187:
5185:
5182:
5180:
5177:
5176:
5174:
5170:
5164:
5161:
5159:
5156:
5154:
5151:
5149:
5146:
5144:
5141:
5139:
5138:Metric tensor
5136:
5134:
5131:
5129:
5126:
5125:
5123:
5119:
5116:
5112:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5050:Exterior form
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5012:
5010:
5004:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4967:
4964:
4962:
4959:
4957:
4954:
4952:
4949:
4947:
4944:
4943:
4941:
4939:
4935:
4929:
4926:
4924:
4923:Tensor bundle
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4880:
4878:
4872:
4866:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4822:
4820:
4816:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4782:
4780:
4775:
4772:
4770:
4767:
4766:
4763:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4718:
4716:
4714:
4710:
4707:
4703:
4699:
4698:
4692:
4688:
4681:
4676:
4674:
4669:
4667:
4662:
4661:
4658:
4646:
4638:
4637:
4634:
4628:
4625:
4623:
4622:Sparse matrix
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4604:
4602:
4600:
4596:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4561:
4559:
4557:constructions
4556:
4552:
4546:
4545:Outermorphism
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4505:Cross product
4503:
4501:
4498:
4497:
4495:
4493:
4489:
4483:
4480:
4478:
4475:
4473:
4472:Outer product
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4452:Orthogonality
4450:
4449:
4447:
4445:
4441:
4435:
4432:
4430:
4429:Cramer's rule
4427:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4399:Decomposition
4397:
4395:
4392:
4391:
4389:
4387:
4383:
4378:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4274:
4272:
4268:
4262:
4259:
4257:
4254:
4253:
4250:
4246:
4239:
4234:
4232:
4227:
4225:
4220:
4219:
4216:
4209:
4205:
4204:
4194:
4192:0-486-42000-0
4188:
4184:
4183:
4177:
4173:
4169:
4165:
4159:
4155:
4151:
4147:
4143:
4139:
4135:
4129:
4125:
4121:
4117:
4113:
4109:
4103:
4099:
4098:
4092:
4088:
4082:
4078:
4074:
4070:
4069:
4064:
4060:
4056:
4050:
4046:
4038:
4034:
4033:
4028:
4024:
4023:
4011:
4006:
3999:
3998:Bourbaki 1989
3994:
3987:
3982:
3975:
3970:
3964:
3963:0-03-010567-6
3960:
3957:
3953:
3949:
3944:
3930:
3926:
3919:
3904:
3903:
3898:
3892:
3877:
3873:
3867:
3859:
3853:
3850:. CRC Press.
3849:
3848:
3840:
3833:
3829:
3825:
3819:
3804:
3800:
3793:
3789:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3760:
3757:
3756:
3750:
3748:
3744:
3739:
3735:
3731:
3721:
3715:
3711:
3707:
3699:
3695:
3691:
3684:
3682:
3678:
3669:
3663:
3657:
3648:
3639:
3635:
3629:
3625:
3601:
3598:
3592:
3586:
3574:
3570:
3566:
3553:
3547:
3544:
3541:
3529:
3525:
3517:
3516:
3515:
3511:
3507:
3503:
3493:
3478:
3477:vector spaces
3474:
3468:
3464:
3460:
3453:
3451:
3445:
3441:
3434:
3430:
3426:
3423:
3422:nondegenerate
3408:
3398:
3396:
3391:
3387:
3383:
3373:
3369:
3365:
3361:
3357:
3353:
3346:
3342:
3338:
3334:
3330:
3326:
3320:
3316:
3312:
3306:
3299:
3295:
3291:
3287:
3283:
3279:
3273:
3269:
3265:
3261:
3255:
3251:
3247:
3240:
3239:Bilinear form
3230:
3228:
3223:
3210:
3205:
3199:
3193:
3188:
3177:
3171:
3160:
3151:
3147:
3136:
3127:
3123:
3118:
3113:
3109:
3105:
3086:
3077:
3073:
3068:
3066:
3061:
3057:
3053:
3047:
3043:
3036:
3022:
3018:
3012:
3008:
3000:
2996:
2992:
2984:
2980:
2976:
2970:
2969:
2968:
2963:
2951:
2950:
2943:
2939:
2933:
2929:
2925:
2919:
2915:
2911:
2907:, is the map
2906:
2902:
2898:
2893:
2889:
2885:
2865:
2857:
2852:
2845:
2835:
2833:
2827:
2825:
2819:
2809:
2807:
2803:
2799:
2795:
2792: ≠
2791:
2787:
2783:
2779:
2775:
2772: ×
2771:
2766:
2764:
2760:
2756:
2751:
2749:
2745:
2741:
2737:
2733:
2726:
2721:
2716:
2692:
2672:
2650:
2635:
2630:
2620:
2598:
2589:
2588:
2587:
2584:
2578:
2575:
2572:
2556:
2549:
2536:
2515:
2509:
2503:
2497:
2491:
2486:
2485:inner product
2481:
2476:
2470:
2466:
2459:
2456:
2449:
2445:
2438:
2433:
2428:
2421:
2417:
2410:
2319:
2293:
2278:
2270:
2254:
2234:
2214:
2150:
2142:
2126:
2106:
2086:
2083:
2080:
2028:
2027:
2025:
2022:
1967:
1966:
1935:
1897:
1889:
1886:
1882:
1877:
1872:
1866:
1862:The notation
1859:
1858:
1845:
1835:
1830:
1827:
1817:
1812:
1807:
1804:
1799:
1784:
1775:
1772:
1767:
1764:
1758:
1753:
1739:
1719:
1711:
1697:
1691:
1686:
1682:
1677:
1673:
1672:
1659:
1642:
1638:
1623:
1612:
1601:
1595:
1589:
1585:
1581:
1577:
1572:
1568:
1563:
1559:
1552:
1546:
1541:
1540:
1538:
1537:
1524:
1497:
1487:
1476:
1467:
1460:
1456:
1448:
1444:
1438:
1434:
1430:
1429:
1416:
1401:
1398:
1388:
1379:
1375:
1366:
1361:
1357:
1356:
1343:
1328:
1313:
1303:
1294:
1285:
1276:
1271:
1267:
1263:
1262:
1249:
1241:
1231:
1216:
1207:
1206:
1205:
1203:
1194:
1188:
1161:
1155:
1150:
1145:
1138:
1133:
1128:
1122:
1117:
1106:
1100:
1095:
1088:
1083:
1076:
1071:
1065:
1055:
1054:
1037:
1031:
1026:
1019:
1014:
1008:
1003:
992:
986:
981:
974:
969:
963:
953:
952:
935:
929:
922:
916:
910:
899:
893:
888:
882:
872:
871:
851:
841:
838:
826:
807:
806:
805:
802:
797:
796:
776:
771:
768:
758:
739:
738:
737:
734:
729:
728:
724:is called an
723:
704:
689:
686:
667:
666:
665:
662:
657:
656:
651:
632:
617:
598:
597:
596:
593:
588:
584:
583:
578:
574:
555:
547:
544:
525:
524:
523:
520:
515:
514:
494:
486:
467:
466:
465:
462:
457:
456:
445:
443:
434:
431:
424:
420:
413:
407:
397:
391:
387:
383:
376:
372:matrix, then
369:
365:
358:
338:
333:
330:
325:
317:
312:
307:
304:
299:
284:
275:
274:
273:
270:
256:
239:
233:
228:
225:
219:
214:
211:
206:
205:main diagonal
201:
197:
194:
193:
192:
189:
183:
177:
171:
152:
148:
138:
132:
126:
122:, denoted by
120:
101:
99:
95:
91:
87:
86:Arthur Cayley
82:
79:
73:
68:
64:
60:
52:
48:
43:
37:
33:
19:
5301:Hermann Weyl
5105:Vector space
5090:Pseudotensor
5055:Fiber bundle
5008:abstractions
4995:
4903:Mixed tensor
4888:Tensor field
4695:
4555:Vector space
4361:
4287:Vector space
4181:
4153:
4119:
4096:
4079:, Springer,
4076:
4073:Halmos, Paul
4036:
4031:
4005:
3993:
3981:
3969:
3951:
3943:
3932:. Retrieved
3928:
3918:
3906:. Retrieved
3900:
3891:
3879:. Retrieved
3875:
3866:
3846:
3839:
3831:
3827:
3818:
3808:September 8,
3806:. Retrieved
3803:Math Insight
3802:
3792:
3740:
3733:
3729:
3713:
3709:
3705:
3702:is equal to
3697:
3693:
3689:
3685:
3667:
3661:
3655:
3644:
3637:
3633:
3627:
3623:
3509:
3505:
3501:
3491:
3486:, we define
3466:
3462:
3458:
3454:
3449:
3443:
3439:
3432:
3428:
3411:
3389:
3385:
3382:homomorphism
3371:
3367:
3363:
3359:
3355:
3351:
3344:
3340:
3336:
3332:
3328:
3324:
3318:
3314:
3310:
3304:
3297:
3293:
3289:
3285:
3281:
3277:
3271:
3267:
3263:
3259:
3253:
3249:
3245:
3242:
3221:
3203:
3200:
3191:
3186:
3175:
3169:
3158:
3149:
3145:
3134:
3125:
3121:
3116:
3111:
3107:
3103:
3084:
3069:
3062:
3055:
3051:
3045:
3041:
3026:
3020:
3016:
3010:
3006:
2998:
2994:
2990:
2982:
2978:
2974:
2947:
2941:
2937:
2931:
2927:
2923:
2917:
2913:
2909:
2904:
2900:
2891:
2887:
2883:
2850:
2847:
2828:
2821:
2793:
2789:
2785:
2773:
2769:
2767:
2752:
2729:
2582:
2579:
2573:
2570:
2554:
2547:
2538:. The entry
2534:
2513:
2507:
2501:
2495:
2489:
2487:of a row of
2479:
2468:
2464:
2457:
2454:
2447:
2443:
2436:
2426:
2419:
2415:
2408:
2405:
1875:
1864:
1765:
1762:
1756:
1695:
1689:
1599:
1593:
1587:
1583:
1579:
1575:
1570:
1566:
1561:
1557:
1550:
1544:
1458:
1454:
1446:
1442:
1192:
1186:
1183:
800:
793:
791:
732:
725:
719:
660:
653:
647:
591:
589:); that is,
580:
570:
518:
511:
509:
460:
453:
451:
435:
429:
422:
418:
411:
405:
395:
392:
385:
381:
374:
367:
363:
356:
353:
268:
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243:
237:
231:
223:
217:
209:
199:
187:
181:
175:
169:
136:
130:
124:
118:
115:
83:
77:
71:
62:
56:
50:
49:of a matrix
46:
5241:Élie Cartan
5189:Spin tensor
5163:Weyl tensor
5121:Mathematics
5085:Multivector
4876:definitions
4774:Engineering
4713:Mathematics
4535:Multivector
4500:Determinant
4457:Dot product
4302:Linear span
4010:Trèves 2006
3986:Halmos 1974
3956:Brooks/Cole
3925:"Transpose"
3681:orthonormal
3647:isomorphism
3395:double dual
3184:. The map
2921:defined by
2899:, then its
2854:denote the
2798:permutation
2424:matrix and
1881:eigenvalues
1685:dot product
1676:determinant
798:; that is,
730:; that is,
658:; that is,
516:; that is,
458:; that is,
427:, notation
5316:Categories
5070:Linear map
4938:Operations
4569:Direct sum
4404:Invertible
4307:Linear map
3934:2020-09-08
3784:References
3514:satisfies
3473:linear map
3227:dual bases
3133:) ⊆
2897:linear map
2816:See also:
2746:, such as
2713:See also:
1433:linear map
1266:involution
1180:Properties
112:Definition
5209:EM tensor
5045:Dimension
4996:Transpose
4599:Numerical
4362:Transpose
4172:853623322
4152:(2006) .
4142:840278135
4029:(1989) .
4000:, II §2.5
3876:ProofWiki
3393:into the
3376:. Here,
3192:transpose
3050: :=
3001:)⟩
2084:×
1828:−
1805:−
1712:⋅
1463:matrices.
1435:from the
847:¯
839:−
769:−
700:¯
690:−
648:A square
628:¯
571:A square
548:−
390:matrix.
262:-th row,
248:-th row,
203:over its
153:⊺
63:transpose
5322:Matrices
5075:Manifold
5060:Geodesic
4818:Notation
4645:Category
4584:Subspace
4579:Quotient
4530:Bivector
4444:Bilinear
4386:Matrices
4261:Glossary
4075:(1974),
4063:18588156
3753:See also
3708: :
3692: :
3649:between
3636:∈
3626:∈
3621:for all
3504: :
3475:between
3461: :
3313: :
3262: :
3248: :
3157:→
3148: :
3106: :
3048:⟩
3040:⟨
3004:for all
2989:⟨
2985:⟩
2973:⟨
2962:relation
2949:pullback
2916:→
2912: :
2886: :
2834:choice.
2778:in-place
2732:computer
2402:Products
2119:and let
1360:addition
868:Examples
442:variable
5172:Physics
5006:Related
4769:Physics
4687:Tensors
4256:Outline
3950:(2006)
3492:adjoint
3490:as the
3450:adjoint
3401:Adjoint
3033:is the
2869:. Let
2780:, with
2776:matrix
2267:is the
1934:similar
1270:inverse
722:inverse
650:complex
573:complex
258:is the
196:Reflect
61:, the
5100:Vector
5095:Spinor
5080:Matrix
4874:Tensor
4540:Tensor
4352:Kernel
4282:Vector
4277:Scalar
4189:
4170:
4160:
4140:
4130:
4104:
4083:
4061:
4051:
4039:]
3961:
3854:
3093:. If
3078:(TVS)
3027:where
2864:module
2858:of an
2788:. For
2736:memory
2412:is an
1268:(self-
1202:scalar
444:name.
378:is an
360:is an
67:matrix
5020:Basis
4705:Scope
4409:Minor
4394:Block
4332:Basis
4041:(PDF)
4035:[
3988:, §44
3908:4 Feb
3881:4 Feb
3677:bases
3471:is a
3322:i.e.
3209:bases
3180:'
3163:'
3154:'
3139:'
3130:'
3089:'
3074:of a
3065:below
2895:is a
2832:basis
2730:On a
2517:. If
2163:. If
2139:be a
1769:is a
1437:space
1200:be a
440:as a
421:) = (
65:of a
4564:Dual
4419:Rank
4187:ISBN
4168:OCLC
4158:ISBN
4138:OCLC
4128:ISBN
4102:ISBN
4081:ISBN
4059:OCLC
4049:ISBN
3959:ISBN
3910:2021
3883:2021
3852:ISBN
3810:2020
3679:are
3665:and
3653:and
3631:and
3482:and
3437:and
3416:and
3362:) =
3335:) =
3288:) =
3215:and
3097:and
3070:The
3014:and
2905:dual
2873:and
2848:Let
2782:O(1)
2748:BLAS
2742:for
2528:and
2452:and
2185:and
2051:and
2029:Let
1990:and
1693:and
1683:The
1674:The
1190:and
1184:Let
4124:GTM
3832:148
3498:if
3494:of
3455:If
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3339:(Ψ(
3211:of
3198:.
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2981:),
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2565:i j
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2073:be
1936:to
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2673:=
2668:T
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2656:T
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2646:T
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2636:(
2631:=
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2615:T
2610:A
2604:A
2599:(
2574:A
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2563:p
2555:p
2548:p
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2329:A
2303:A
2291:.
2279:k
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2235:k
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1107:]
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970:1
964:[
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759:=
754:T
749:A
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705:.
696:A
687:=
682:T
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633:.
624:A
618:=
613:T
608:A
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545:=
540:T
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495:.
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487:=
482:T
477:A
461:A
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425:)
423:A
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