5580:
3570:
20:
5844:
555:
875:
305:
597:
550:{\displaystyle {\begin{bmatrix}2&3&1\\0&8&-2\end{bmatrix}}\circ {\begin{bmatrix}3&1&4\\7&9&5\end{bmatrix}}={\begin{bmatrix}2\times 3&3\times 1&1\times 4\\0\times 7&8\times 9&-2\times 5\end{bmatrix}}={\begin{bmatrix}6&3&4\\0&72&-10\end{bmatrix}}}
1063:
1988:
1182:
3030:
4156:
1775:
1327:
870:{\displaystyle {\begin{aligned}A\odot B&=B\odot A,\\A\odot (B\odot C)&=(A\odot B)\odot C,\\A\odot (B+C)&=A\odot B+A\odot C,\\\left(kA\right)\odot B&=A\odot \left(kB\right)=k\left(A\odot B\right),\\A\odot 0&=0\odot A=0.\end{aligned}}}
3260:
1416:
1499:
2108:
3330:
which can apply any function element-wise. This includes both binary operators (such as the aforementioned multiplication and exponentiation, as well as any other binary operator such as the
Kronecker product), and also unary operators such as
1620:
3144:
2918:
2199:
947:
4506:
2557:
2362:
906:
under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. Furthermore, a matrix has an inverse under
Hadamard multiplication if and only if none of the elements are equal to
1825:
2925:
3849:
2794:
4591:
1643:
1201:
2663:
3166:
3050:
2930:
2830:
1830:
1098:
602:
3161:
1332:
251:
1421:
2043:
4196:
4000:
1554:
1531:
3045:
2825:
2131:
55:
of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the
3543:, the Hadamard operator can be used for enhancing, suppressing or masking image regions. One matrix represents the original image, the other acts as weight or masking matrix.
2138:
5251:
Sak, Haşim; Senior, Andrew; Beaufays, Françoise (2014-02-05). "Long Short-Term Memory Based
Recurrent Neural Network Architectures for Large Vocabulary Speech Recognition".
4423:
4355:
3837:
4613:
2221:
2034:
2012:
5211:
4529:
2577:
2382:
162:
136:
2683:
3688:
3710:
3635:
3609:
2473:
2278:
2466:
2446:
2426:
2406:
2261:
2241:
4950:
Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz
Neudecker and matrix differential calculus".
4429:
5406:
1058:{\displaystyle \mathbf {x} ^{*}(A\circ B)\mathbf {y} =\operatorname {tr} \left({D}_{\mathbf {x} }^{*}A{D}_{\mathbf {y} }{B}^{\mathsf {T}}\right),}
2726:
2588:
5094:"Supplementary Material: Tensor Displays: Compressive Light Field Synthesis using Multilayer Displays with Directional Backlighting"
5438:
5771:
3746:
5829:
5272:
Neudecker, Heinz; Liu, Shuangzhe; Polasek, Wolfgang (1995). "The
Hadamard product and some of its applications in statistics".
1983:{\displaystyle {\begin{aligned}D(A\odot B)E&=(DAE)\odot B=(DA)\odot (BE)\\&=(AE)\odot (DB)=A\odot (DBE).\end{aligned}}}
4541:
1177:{\displaystyle \operatorname {tr} \left(AB^{\mathsf {T}}\right)=\mathbf {1} ^{\mathsf {T}}\left(A\odot B\right)\mathbf {1} }
170:
3025:{\displaystyle {\begin{aligned}{B}&={A}^{\circ {\frac {1}{2}}}\\B_{ij}&={A_{ij}}^{\frac {1}{2}}\end{aligned}}}
2580:
5169:
5125:
5155:
5819:
4165:
4151:{\displaystyle {A}={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}},\quad {B}=\left=\left}
3941:
3715:
5781:
5717:
5093:
5299:
Neudecker, Heinz; Liu, Shuangzhe (2001). "Some statistical properties of
Hadamard products of random matrices".
1770:{\displaystyle \prod _{i=k}^{n}\lambda _{i}(A\odot B)\geq \prod _{i=k}^{n}\lambda _{i}(AB),\quad k=1,\ldots ,n,}
23:
The
Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions.
5225:
1322:{\displaystyle \sum _{i}(A\circ B)_{ij}=\left(B^{\mathsf {T}}A\right)_{jj}=\left(AB^{\mathsf {T}}\right)_{ii}.}
1081:. In particular, using vectors of ones, this shows that the sum of all elements in the Hadamard product is the
1504:
3353:
5559:
5431:
3308:
3255:{\displaystyle {\begin{aligned}{C}&={A}\oslash {B}\\C_{ij}&={\frac {A_{ij}}{B_{ij}}}\end{aligned}}}
2116:
1411:{\displaystyle \left(\mathbf {y} \mathbf {x} ^{*}\right)\odot A={D}_{\mathbf {y} }A{D}_{\mathbf {x} }^{*}}
5514:
1494:{\displaystyle (A\odot B)\mathbf {y} =\operatorname {diag} \left(AD_{\mathbf {y} }B^{\mathsf {T}}\right)}
2103:{\displaystyle \mathbf {a} \odot \mathbf {b} =D_{\mathbf {a} }\mathbf {b} =D_{\mathbf {b} }\mathbf {a} }
5569:
5463:
5006:
4825:
4739:
3469:
5809:
5458:
4911:"Matrix differential calculus with applications in the multivariate linear model and its diagnostics"
4633:
3428:
5190:
4890:
Liu, Shuangzhe; Trenkler, Götz (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products".
4367:
3513:
2819:(which are in effect the same thing because of fractional indices), defined for a matrix such that:
1615:{\displaystyle \operatorname {rank} (A\odot B)\leq \operatorname {rank} (A)\operatorname {rank} (B)}
5801:
5684:
3473:
3465:
2723:
of their
Hadamard product is greater than or equal to the product of their respective determinants:
2704:
1637:
4596:
2707:
is positive-semidefinite. This is known as the Schur product theorem, after
Russian mathematician
2204:
2017:
1995:
5847:
5776:
5554:
5424:
4645:
3139:{\displaystyle {\begin{aligned}{B}&={A}^{\circ -1}\\B_{ij}&={A_{ij}}^{-1}\end{aligned}}}
5868:
5611:
5544:
5534:
4660:
4532:
3555:
2686:
2113:
1082:
5115:
4514:
3536:. The decoding step involves an entry-for-entry product, in other words the Hadamard product.
2913:{\displaystyle {\begin{aligned}{B}&={A}^{\circ 2}\\B_{ij}&={A_{ij}}^{2}\end{aligned}}}
2562:
2367:
141:
115:
5626:
5621:
5616:
5549:
5494:
5400:
4629:
3412:
3270:
2698:
575:
75:
56:
3319:, and other operators are analogously defined element-wise, for example Hadamard powers use
5873:
5636:
5601:
5588:
5479:
4909:
Liu, Shuangzhe; Leiva, Víctor; Zhuang, Dan; Ma, Tiefeng; Figueroa-Zúñiga, Jorge I. (2022).
3551:
3392:
will produce the matrix product. The
Hadamard product can be obtained with the method call
2668:
52:
3652:
8:
5814:
5694:
5669:
5519:
3693:
3618:
3592:
3356:
does not have built-in array support, leading to inconsistent/conflicting notations. The
2194:{\displaystyle \operatorname {diag} (\mathbf {a} )=(\mathbf {a} \mathbf {1} ^{T})\odot I}
1072:
5524:
5386:
5363:
5316:
5252:
5141:
4932:
4854:
Styan, George P. H. (1973), "Hadamard Products and Multivariate Statistical Analysis",
4715:
2451:
2431:
2411:
2391:
2246:
2226:
5077:
5060:
4767:
3550:
literature, for example, to describe the architecture of recurrent neural networks as
5722:
5679:
5606:
5499:
5383:
The International Conference on Learning Representations (ICLR) 2017. – Toulon, 2017.
5367:
5320:
5121:
4936:
4867:
4719:
4655:
4650:
4501:{\displaystyle {M}\bullet {M}={M}\left({M}\otimes \mathbf {1} ^{\textsf {T}}\right),}
3529:
3401:
2385:
1819:
1545:
3561:
It is also used to study the statistical properties of random vectors and matrices.
5727:
5631:
5484:
5355:
5308:
5281:
5072:
4986:
4955:
4922:
4871:
4863:
4707:
3547:
3540:
3477:
3447:
2803:
Other Hadamard operations are also seen in the mathematical literature, namely the
1092:
60:
48:
5786:
5579:
5539:
5529:
4803:
3340:
2264:
2037:
903:
299:
For example, the Hadamard product for two arbitrary 2 × 3 matrices is:
3288:. It also has analogous dot operators which include, for example, the operators
5791:
5712:
5447:
4959:
4927:
4910:
890:
5285:
5061:"Hadamard inverses, square roots and products of almost semidefinite matrices"
4991:
4974:
4876:
3396:. Some Python packages include support for Hadamard powers using methods like
3323:. But unlike MATLAB, in Julia this "dot" syntax is generalized with a generic
1192:, is that the row-sums of their Hadamard product are the diagonal elements of
5862:
5824:
5747:
5707:
5674:
5654:
3578:
5348:
Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999
2552:{\displaystyle (A\bullet B)\odot (C\bullet D)=(A\odot C)\bullet (B\odot D),}
2357:{\displaystyle (A\otimes B)\odot (C\otimes D)=(A\odot C)\otimes (B\odot D),}
164:) is a matrix of the same dimension as the operands, with elements given by
5757:
5646:
5596:
5489:
5031:
3569:
5312:
4785:
944:
with these vectors as their main diagonals, the following identity holds:
5737:
5702:
5659:
5504:
3481:
2720:
2708:
571:
567:
79:
71:
64:
28:
3492:
apply the Hadamard product, whereas the matrix product is written using
19:
5766:
5509:
5359:
5117:
Object Detection and Recognition in Digital Images: Theory and Practice
4711:
1798:
3311:
has similar syntax as MATLAB, where Hadamard multiplication is called
1640:, then the following inequality involving the Hadamard product holds:
5564:
5092:
Wetzstein, Gordon; Lanman, Douglas; Hirsch, Matthew; Raskar, Ramesh.
1541:
59:. It is attributed to, and named after, either French mathematician
5732:
5391:
4698:
Davis, Chandler (1962). "The norm of the Schur product operation".
3451:
1418:
Furthermore, a Hadamard matrix-vector product can be expressed as:
5340:
5257:
5416:
3461:
3304:
for matrix multiplication and matrix exponentials, respectively.
5742:
4625:
3277:
3377:
3357:
4975:"On an eigenvalue inequality involving the Hadamard product"
3533:
5341:"A Family of Face Products of Matrices and its properties"
2789:{\displaystyle \det({A}\odot {B})\geq \det({A})\det({B}).}
2036:
is the same as matrix multiplication of the corresponding
4949:
4892:
International Journal of Information and Systems Sciences
879:
The identity matrix under Hadamard multiplication of two
5380:
4586:{\displaystyle \mathbf {c} \bullet {M}=\mathbf {c} {M},}
5091:
3296:. Because of this mechanism, it is possible to reserve
3280:, the Hadamard product is expressed as "dot multiply":
5232:. The R Project for Statistical Computing. 16 May 2013
3866:
503:
415:
366:
314:
4599:
4544:
4517:
4432:
4370:
4168:
3852:
3718:
3696:
3655:
3621:
3595:
3520:
for Hadamard Product of numeric matrices or vectors.
3164:
3048:
2928:
2828:
2729:
2671:
2658:{\displaystyle (A\bullet B)(C\ast D)=(AC)\odot (BD),}
2591:
2565:
2476:
2454:
2434:
2414:
2394:
2370:
2281:
2249:
2229:
2207:
2141:
2119:
2046:
2020:
1998:
1828:
1646:
1557:
1507:
1424:
1335:
1204:
1101:
950:
600:
308:
173:
144:
118:
4786:"linear algebra - What does a dot in a circle mean?"
5271:
3273:include the Hadamard product, under various names.
1551:The Hadamard product satisfies the rank inequality
5381:Ha D., Dai A.M., Le Q.V. (2017). "HyperNetworks".
4804:"Element-wise (or pointwise) operations notation?"
4607:
4585:
4523:
4500:
4417:
4349:
4150:
3831:
3704:
3682:
3629:
3603:
3254:
3138:
3024:
2912:
2788:
2677:
2657:
2571:
2551:
2460:
2440:
2420:
2400:
2376:
2356:
2255:
2235:
2215:
2193:
2125:
2102:
2028:
2006:
1982:
1769:
1614:
1533:is the vector formed from the diagonals of matrix
1525:
1493:
1410:
1321:
1176:
1057:
869:
549:
245:
156:
130:
5250:
246:{\displaystyle (A\odot B)_{ij}=(A)_{ij}(B)_{ij}.}
5860:
5032:"End products in matrices in radar applications"
4908:
2769:
2755:
2730:
2135:may be expressed using the Hadamard product as:
3564:
4819:
4817:
4768:"Hadamard product - Machine Learning Glossary"
5432:
4678:
4676:
2271:
5405:: CS1 maint: multiple names: authors list (
5298:
4683:Horn, Roger A.; Johnson, Charles R. (2012).
4624:The penetrating face product is used in the
5334:
5332:
5330:
5212:"Common Matrices — SymPy 1.9 documentation"
5039:Radioelectronics and Communications Systems
4973:Hiai, Fumio; Lin, Minghua (February 2017).
4889:
4849:
4847:
4845:
4814:
4682:
4636:models, specifically convolutional layers.
3264:
5439:
5425:
4673:
5390:
5256:
5076:
4990:
4926:
4875:
4484:
2711:. For two positive-semidefinite matrices
5327:
5054:
5052:
4842:
3573:The penetrating face product of matrices
3568:
2692:
1526:{\displaystyle \operatorname {diag} (M)}
900:matrix where all elements are equal to 1
570:(when working with a commutative ring),
78:. Unlike the matrix product, it is also
18:
5338:
5113:
5029:
5023:
4972:
4737:
2798:
5861:
5830:Comparison of linear algebra libraries
5191:"Dot Syntax for Vectorizing Functions"
5120:. John Wiley & Sons. p. 109.
1480:
1296:
1255:
1144:
1122:
1041:
922:, and corresponding diagonal matrices
296:), the Hadamard product is undefined.
256:For matrices of different dimensions (
5420:
5374:
5058:
5049:
4853:
4738:Million, Elizabeth (April 12, 2007).
4697:
4632:. This operation can also be used in
2126:{\displaystyle \operatorname {diag} }
1992:The Hadamard product of two vectors
4733:
4731:
4729:
3581:the penetrating face product of the
3380:symbolic library, multiplication of
1540:The Hadamard product is a principal
5065:Linear Algebra and Its Applications
4979:Linear Algebra and Its Applications
4856:Linear Algebra and Its Applications
4823:
2223:is a constant vector with elements
2040:of one vector by the other vector:
590:are matrices of the same size, and
16:Elementwise product of two matrices
13:
5446:
4359:
3368:as the Hadamard product, and uses
14:
5885:
5339:Slyusar, V. I. (March 13, 1998).
4726:
4619:
3376:for the matrix product. With the
5843:
5842:
5820:Basic Linear Algebra Subprograms
5578:
4915:Journal of Multivariate Analysis
4601:
4562:
4546:
4478:
3528:The Hadamard product appears in
2209:
2172:
2166:
2152:
2096:
2089:
2076:
2069:
2056:
2048:
2022:
2000:
1468:
1441:
1397:
1380:
1348:
1342:
1170:
1138:
1091:where superscript T denotes the
1027:
1005:
979:
953:
5718:Seven-dimensional cross product
5292:
5265:
5244:
5218:
5204:
5183:
5162:
5148:
5134:
5107:
5085:
4999:
4966:
4943:
4902:
3927:
3577:According to the definition of
3523:
3484:), the multiplication operator
1742:
4883:
4796:
4778:
4760:
4691:
4572:
4566:
4460:
4454:
4418:{\displaystyle {A}{B}={B}{A};}
4404:
4398:
4382:
4376:
4180:
4174:
3730:
3724:
3677:
3664:
3512:, respectively. The R package
2780:
2772:
2766:
2758:
2749:
2733:
2705:positive-semidefinite matrices
2649:
2640:
2634:
2625:
2619:
2607:
2604:
2592:
2543:
2531:
2525:
2513:
2507:
2495:
2489:
2477:
2348:
2336:
2330:
2318:
2312:
2300:
2294:
2282:
2182:
2162:
2156:
2148:
2112:The vector to diagonal matrix
1970:
1958:
1946:
1937:
1931:
1922:
1909:
1900:
1894:
1885:
1873:
1861:
1848:
1836:
1736:
1727:
1690:
1678:
1609:
1603:
1594:
1588:
1576:
1564:
1520:
1514:
1437:
1425:
1228:
1215:
1184:. A related result for square
975:
963:
711:
699:
677:
665:
655:
643:
228:
221:
209:
202:
187:
174:
1:
5156:"Array vs. Matrix Operations"
5078:10.1016/S0024-3795(98)10162-3
4687:. Cambridge University Press.
4666:
4350:{\displaystyle {A}{B}=\left.}
3832:{\displaystyle {A}{B}=\left.}
3435:to make compact expressions (
3360:numerical library interprets
3269:Most scientific or numerical
902:. This is different from the
560:
85:
5560:Eigenvalues and eigenvectors
5170:"Vectorized "dot" operators"
4868:10.1016/0024-3795(73)90023-2
4608:{\displaystyle \mathbf {c} }
3565:The penetrating face product
3454:, the operation is known as
2719:, it is also known that the
2703:The Hadamard product of two
2216:{\displaystyle \mathbf {1} }
2029:{\displaystyle \mathbf {b} }
2007:{\displaystyle \mathbf {a} }
7:
4639:
2408:has the same dimensions of
578:over addition. That is, if
10:
5890:
5114:Cyganek, Boguslaw (2013).
4960:10.1007/s00362-023-01499-w
4928:10.1016/j.jmva.2021.104849
4808:Mathematics Stack Exchange
4790:Mathematics Stack Exchange
3841:
3431:library uses the operator
2696:
2272:The mixed-product property
1638:positive-definite matrices
5838:
5800:
5756:
5693:
5645:
5587:
5576:
5472:
5454:
5286:10.1080/02331889508802503
4992:10.1016/j.laa.2016.11.017
4634:artificial neural network
3394:a.multiply_elementwise(b)
3307:The programming language
4524:{\displaystyle \bullet }
3516:introduces the function
3419:member function for the
3339:. Thus, any function in
3313:broadcast multiplication
3284:, or the function call:
3265:In programming languages
2572:{\displaystyle \bullet }
2377:{\displaystyle \otimes }
566:The Hadamard product is
157:{\displaystyle A\circ B}
131:{\displaystyle A\odot B}
70:The Hadamard product is
63:or German mathematician
5226:"Matrix multiplication"
5142:"MATLAB times function"
5030:Slyusar, V. I. (1998).
5012:. buzzard.ups.edu. 2007
4646:Frobenius inner product
112:, the Hadamard product
5545:Row and column vectors
5059:Reams, Robert (1999).
4826:"The Hadamard Product"
4740:"The Hadamard Product"
4630:digital antenna arrays
4609:
4587:
4533:face-splitting product
4525:
4502:
4419:
4351:
4152:
3833:
3706:
3690:) is a matrix of size
3684:
3631:
3605:
3574:
3443:is a matrix product).
3325:broadcasting operator
3315:and also denoted with
3256:
3140:
3026:
2914:
2790:
2679:
2659:
2581:face-splitting product
2573:
2553:
2462:
2442:
2422:
2402:
2378:
2358:
2257:
2237:
2217:
2195:
2127:
2104:
2030:
2008:
1984:
1771:
1716:
1667:
1616:
1527:
1495:
1412:
1323:
1178:
1059:
871:
551:
247:
158:
132:
102:of the same dimension
24:
5550:Row and column spaces
5495:Scalar multiplication
5313:10.1007/s003620100074
4700:Numerische Mathematik
4610:
4588:
4526:
4503:
4420:
4352:
4153:
3834:
3707:
3685:
3632:
3606:
3572:
3456:array multiplication.
3271:programming languages
3257:
3141:
3027:
2915:
2791:
2699:Schur product theorem
2693:Schur product theorem
2685: is column-wise
2680:
2678:{\displaystyle \ast }
2660:
2574:
2554:
2463:
2443:
2423:
2403:
2379:
2359:
2258:
2238:
2218:
2196:
2128:
2105:
2031:
2009:
1985:
1772:
1696:
1647:
1617:
1528:
1496:
1413:
1324:
1179:
1060:
872:
552:
248:
159:
133:
22:
5685:Gram–Schmidt process
5637:Gaussian elimination
5230:An Introduction to R
4824:Million, Elizabeth.
4597:
4542:
4515:
4430:
4368:
4166:
3850:
3716:
3694:
3683:{\displaystyle {B}=}
3653:
3619:
3615:-dimensional matrix
3593:
3162:
3046:
2926:
2826:
2799:Analogous operations
2727:
2669:
2589:
2563:
2474:
2452:
2432:
2412:
2392:
2368:
2279:
2247:
2227:
2205:
2139:
2117:
2044:
2018:
1996:
1826:
1644:
1555:
1505:
1422:
1333:
1202:
1099:
948:
598:
306:
171:
142:
116:
37:element-wise product
5815:Numerical stability
5695:Multilinear algebra
5670:Inner product space
5520:Linear independence
4772:machinelearning.wtf
3705:{\displaystyle {B}}
3630:{\displaystyle {B}}
3604:{\displaystyle {A}}
3532:algorithms such as
3415:library provides a
1407:
1073:conjugate transpose
1015:
35:(also known as the
5525:Linear combination
5360:10.1007/BF02733426
5301:Statistical Papers
4952:Statistical Papers
4877:10338.dmlcz/102190
4712:10.1007/bf01386329
4661:Khatri–Rao product
4628:-matrix theory of
4605:
4583:
4521:
4498:
4415:
4347:
4338:
4148:
4142:
3986:
3918:
3829:
3820:
3702:
3680:
3627:
3601:
3575:
3546:It is used in the
3384:objects as either
3346:can be applied as
3252:
3250:
3136:
3134:
3022:
3020:
2910:
2908:
2786:
2687:Khatri–Rao product
2675:
2655:
2569:
2549:
2458:
2438:
2418:
2398:
2374:
2354:
2253:
2233:
2213:
2191:
2123:
2100:
2026:
2004:
1980:
1978:
1767:
1612:
1523:
1491:
1408:
1389:
1319:
1214:
1174:
1055:
997:
867:
865:
547:
541:
489:
401:
352:
243:
154:
128:
51:that takes in two
25:
5856:
5855:
5723:Geometric algebra
5680:Kronecker product
5515:Linear projection
5500:Vector projection
4656:Kronecker product
4651:Pointwise product
4486:
3530:lossy compression
3425:a.cwiseProduct(b)
3246:
3152:Hadamard division
3015:
2963:
2461:{\displaystyle D}
2441:{\displaystyle B}
2421:{\displaystyle C}
2401:{\displaystyle A}
2386:Kronecker product
2256:{\displaystyle I}
2236:{\displaystyle 1}
1820:diagonal matrices
1546:Kronecker product
1205:
90:For two matrices
41:entrywise product
5881:
5846:
5845:
5728:Exterior algebra
5665:Hadamard product
5582:
5570:Linear equations
5441:
5434:
5427:
5418:
5417:
5411:
5410:
5404:
5396:
5394:
5378:
5372:
5371:
5345:
5336:
5325:
5324:
5296:
5290:
5289:
5269:
5263:
5262:
5260:
5248:
5242:
5241:
5239:
5237:
5222:
5216:
5215:
5208:
5202:
5201:
5199:
5197:
5187:
5181:
5180:
5178:
5176:
5166:
5160:
5159:
5152:
5146:
5145:
5138:
5132:
5131:
5111:
5105:
5104:
5098:
5089:
5083:
5082:
5080:
5056:
5047:
5046:
5036:
5027:
5021:
5020:
5018:
5017:
5011:
5003:
4997:
4996:
4994:
4970:
4964:
4963:
4947:
4941:
4940:
4930:
4906:
4900:
4899:
4887:
4881:
4880:
4879:
4851:
4840:
4839:
4837:
4835:
4830:
4821:
4812:
4811:
4800:
4794:
4793:
4782:
4776:
4775:
4764:
4758:
4757:
4755:
4753:
4744:
4735:
4724:
4723:
4695:
4689:
4688:
4680:
4614:
4612:
4611:
4606:
4604:
4592:
4590:
4589:
4584:
4579:
4565:
4557:
4549:
4530:
4528:
4527:
4522:
4507:
4505:
4504:
4499:
4494:
4490:
4489:
4488:
4487:
4481:
4472:
4453:
4445:
4437:
4424:
4422:
4421:
4416:
4411:
4397:
4389:
4375:
4356:
4354:
4353:
4348:
4343:
4339:
4187:
4173:
4157:
4155:
4154:
4149:
4147:
4143:
3991:
3987:
3983:
3982:
3977:
3969:
3968:
3963:
3955:
3954:
3949:
3932:
3923:
3922:
3857:
3838:
3836:
3835:
3830:
3825:
3821:
3817:
3816:
3811:
3802:
3790:
3789:
3784:
3775:
3768:
3767:
3762:
3753:
3737:
3723:
3711:
3709:
3708:
3703:
3701:
3689:
3687:
3686:
3681:
3676:
3675:
3660:
3636:
3634:
3633:
3628:
3626:
3610:
3608:
3607:
3602:
3600:
3548:machine learning
3541:image processing
3519:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3478:Wolfram Language
3442:
3438:
3434:
3426:
3422:
3418:
3407:
3399:
3395:
3391:
3387:
3383:
3375:
3371:
3367:
3363:
3349:
3345:
3338:
3334:
3328:
3322:
3318:
3303:
3299:
3295:
3291:
3287:
3283:
3261:
3259:
3258:
3253:
3251:
3247:
3245:
3244:
3232:
3231:
3219:
3210:
3209:
3193:
3185:
3173:
3154:
3153:
3145:
3143:
3142:
3137:
3135:
3131:
3130:
3122:
3121:
3120:
3099:
3098:
3082:
3081:
3070:
3057:
3040:
3039:
3038:Hadamard inverse
3031:
3029:
3028:
3023:
3021:
3017:
3016:
3008:
3006:
3005:
3004:
2983:
2982:
2966:
2965:
2964:
2956:
2950:
2937:
2919:
2917:
2916:
2911:
2909:
2905:
2904:
2899:
2898:
2897:
2876:
2875:
2859:
2858:
2850:
2837:
2817:
2816:
2809:
2808:
2795:
2793:
2792:
2787:
2779:
2765:
2748:
2740:
2718:
2714:
2684:
2682:
2681:
2676:
2664:
2662:
2661:
2656:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2467:
2465:
2464:
2459:
2447:
2445:
2444:
2439:
2427:
2425:
2424:
2419:
2407:
2405:
2404:
2399:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2262:
2260:
2259:
2254:
2242:
2240:
2239:
2234:
2222:
2220:
2219:
2214:
2212:
2200:
2198:
2197:
2192:
2181:
2180:
2175:
2169:
2155:
2132:
2130:
2129:
2124:
2109:
2107:
2106:
2101:
2099:
2094:
2093:
2092:
2079:
2074:
2073:
2072:
2059:
2051:
2035:
2033:
2032:
2027:
2025:
2013:
2011:
2010:
2005:
2003:
1989:
1987:
1986:
1981:
1979:
1915:
1817:
1813:
1806:
1796:
1790:
1776:
1774:
1773:
1768:
1726:
1725:
1715:
1710:
1677:
1676:
1666:
1661:
1635:
1629:
1621:
1619:
1618:
1613:
1536:
1532:
1530:
1529:
1524:
1500:
1498:
1497:
1492:
1490:
1486:
1485:
1484:
1483:
1473:
1472:
1471:
1444:
1417:
1415:
1414:
1409:
1406:
1401:
1400:
1394:
1385:
1384:
1383:
1377:
1362:
1358:
1357:
1356:
1351:
1345:
1328:
1326:
1325:
1320:
1315:
1314:
1306:
1302:
1301:
1300:
1299:
1277:
1276:
1268:
1264:
1260:
1259:
1258:
1239:
1238:
1213:
1197:
1191:
1187:
1183:
1181:
1180:
1175:
1173:
1168:
1164:
1149:
1148:
1147:
1141:
1132:
1128:
1127:
1126:
1125:
1093:matrix transpose
1090:
1080:
1070:
1064:
1062:
1061:
1056:
1051:
1047:
1046:
1045:
1044:
1038:
1032:
1031:
1030:
1024:
1014:
1009:
1008:
1002:
982:
962:
961:
956:
943:
932:
921:
915:
899:
888:
876:
874:
873:
868:
866:
824:
820:
799:
795:
764:
760:
556:
554:
553:
548:
546:
545:
494:
493:
406:
405:
357:
356:
295:
285:
275:
265:
252:
250:
249:
244:
239:
238:
220:
219:
198:
197:
163:
161:
160:
155:
137:
135:
134:
129:
111:
101:
95:
61:Jacques Hadamard
49:binary operation
33:Hadamard product
5889:
5888:
5884:
5883:
5882:
5880:
5879:
5878:
5859:
5858:
5857:
5852:
5834:
5796:
5752:
5689:
5641:
5583:
5574:
5540:Change of basis
5530:Multilinear map
5468:
5450:
5445:
5415:
5414:
5398:
5397:
5379:
5375:
5343:
5337:
5328:
5297:
5293:
5270:
5266:
5249:
5245:
5235:
5233:
5224:
5223:
5219:
5210:
5209:
5205:
5195:
5193:
5189:
5188:
5184:
5174:
5172:
5168:
5167:
5163:
5154:
5153:
5149:
5140:
5139:
5135:
5128:
5112:
5108:
5096:
5090:
5086:
5057:
5050:
5034:
5028:
5024:
5015:
5013:
5009:
5005:
5004:
5000:
4971:
4967:
4948:
4944:
4907:
4903:
4888:
4884:
4852:
4843:
4833:
4831:
4828:
4822:
4815:
4802:
4801:
4797:
4784:
4783:
4779:
4766:
4765:
4761:
4751:
4749:
4747:buzzard.ups.edu
4742:
4736:
4727:
4696:
4692:
4685:Matrix analysis
4681:
4674:
4669:
4642:
4622:
4600:
4598:
4595:
4594:
4575:
4561:
4553:
4545:
4543:
4540:
4539:
4516:
4513:
4512:
4483:
4482:
4477:
4476:
4468:
4467:
4463:
4449:
4441:
4433:
4431:
4428:
4427:
4407:
4393:
4385:
4371:
4369:
4366:
4365:
4362:
4360:Main properties
4337:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4290:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4243:
4242:
4237:
4232:
4227:
4222:
4217:
4212:
4207:
4202:
4195:
4191:
4183:
4169:
4167:
4164:
4163:
4141:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4094:
4093:
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4053:
4047:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4011:
4006:
3999:
3995:
3985:
3984:
3978:
3973:
3972:
3970:
3964:
3959:
3958:
3956:
3950:
3945:
3944:
3940:
3936:
3928:
3917:
3916:
3911:
3906:
3900:
3899:
3894:
3889:
3883:
3882:
3877:
3872:
3862:
3861:
3853:
3851:
3848:
3847:
3844:
3819:
3818:
3812:
3807:
3806:
3798:
3796:
3791:
3785:
3780:
3779:
3771:
3769:
3763:
3758:
3757:
3749:
3745:
3741:
3733:
3719:
3717:
3714:
3713:
3697:
3695:
3692:
3691:
3671:
3667:
3656:
3654:
3651:
3650:
3622:
3620:
3617:
3616:
3596:
3594:
3591:
3590:
3567:
3526:
3518:hadamard.prod()
3517:
3509:
3505:
3501:
3497:
3493:
3489:
3485:
3440:
3436:
3432:
3424:
3420:
3416:
3405:
3397:
3393:
3389:
3385:
3381:
3373:
3369:
3365:
3361:
3347:
3343:
3341:prefix notation
3336:
3332:
3326:
3320:
3316:
3301:
3297:
3293:
3289:
3285:
3281:
3267:
3249:
3248:
3237:
3233:
3224:
3220:
3218:
3211:
3202:
3198:
3195:
3194:
3189:
3181:
3174:
3169:
3165:
3163:
3160:
3159:
3156:is defined as:
3151:
3150:
3133:
3132:
3123:
3113:
3109:
3108:
3107:
3100:
3091:
3087:
3084:
3083:
3071:
3066:
3065:
3058:
3053:
3049:
3047:
3044:
3043:
3037:
3036:
3019:
3018:
3007:
2997:
2993:
2992:
2991:
2984:
2975:
2971:
2968:
2967:
2955:
2951:
2946:
2945:
2938:
2933:
2929:
2927:
2924:
2923:
2907:
2906:
2900:
2890:
2886:
2885:
2884:
2877:
2868:
2864:
2861:
2860:
2851:
2846:
2845:
2838:
2833:
2829:
2827:
2824:
2823:
2814:
2813:
2806:
2805:
2801:
2775:
2761:
2744:
2736:
2728:
2725:
2724:
2716:
2712:
2701:
2695:
2670:
2667:
2666:
2590:
2587:
2586:
2564:
2561:
2560:
2475:
2472:
2471:
2453:
2450:
2449:
2433:
2430:
2429:
2413:
2410:
2409:
2393:
2390:
2389:
2369:
2366:
2365:
2280:
2277:
2276:
2274:
2265:identity matrix
2248:
2245:
2244:
2228:
2225:
2224:
2208:
2206:
2203:
2202:
2176:
2171:
2170:
2165:
2151:
2140:
2137:
2136:
2118:
2115:
2114:
2095:
2088:
2087:
2083:
2075:
2068:
2067:
2063:
2055:
2047:
2045:
2042:
2041:
2038:diagonal matrix
2021:
2019:
2016:
2015:
1999:
1997:
1994:
1993:
1977:
1976:
1913:
1912:
1854:
1829:
1827:
1824:
1823:
1815:
1811:
1802:
1792:
1783:
1778:
1721:
1717:
1711:
1700:
1672:
1668:
1662:
1651:
1645:
1642:
1641:
1631:
1625:
1556:
1553:
1552:
1534:
1506:
1503:
1502:
1479:
1478:
1474:
1467:
1466:
1462:
1458:
1454:
1440:
1423:
1420:
1419:
1402:
1396:
1395:
1390:
1379:
1378:
1373:
1372:
1352:
1347:
1346:
1341:
1340:
1336:
1334:
1331:
1330:
1307:
1295:
1294:
1290:
1286:
1282:
1281:
1269:
1254:
1253:
1249:
1248:
1244:
1243:
1231:
1227:
1209:
1203:
1200:
1199:
1193:
1189:
1185:
1169:
1154:
1150:
1143:
1142:
1137:
1136:
1121:
1120:
1116:
1112:
1108:
1100:
1097:
1096:
1086:
1076:
1066:
1040:
1039:
1034:
1033:
1026:
1025:
1020:
1019:
1010:
1004:
1003:
998:
996:
992:
978:
957:
952:
951:
949:
946:
945:
942:
934:
931:
923:
917:
911:
904:identity matrix
891:
889:matrices is an
880:
864:
863:
841:
829:
828:
810:
806:
788:
784:
771:
753:
749:
746:
745:
714:
690:
689:
658:
634:
633:
614:
601:
599:
596:
595:
563:
540:
539:
531:
526:
520:
519:
514:
509:
499:
498:
488:
487:
473:
462:
450:
449:
438:
427:
411:
410:
400:
399:
394:
389:
383:
382:
377:
372:
362:
361:
351:
350:
342:
337:
331:
330:
325:
320:
310:
309:
307:
304:
303:
287:
277:
267:
257:
231:
227:
212:
208:
190:
186:
172:
169:
168:
143:
140:
139:
117:
114:
113:
103:
97:
91:
88:
17:
12:
11:
5:
5887:
5877:
5876:
5871:
5854:
5853:
5851:
5850:
5839:
5836:
5835:
5833:
5832:
5827:
5822:
5817:
5812:
5810:Floating-point
5806:
5804:
5798:
5797:
5795:
5794:
5792:Tensor product
5789:
5784:
5779:
5777:Function space
5774:
5769:
5763:
5761:
5754:
5753:
5751:
5750:
5745:
5740:
5735:
5730:
5725:
5720:
5715:
5713:Triple product
5710:
5705:
5699:
5697:
5691:
5690:
5688:
5687:
5682:
5677:
5672:
5667:
5662:
5657:
5651:
5649:
5643:
5642:
5640:
5639:
5634:
5629:
5627:Transformation
5624:
5619:
5617:Multiplication
5614:
5609:
5604:
5599:
5593:
5591:
5585:
5584:
5577:
5575:
5573:
5572:
5567:
5562:
5557:
5552:
5547:
5542:
5537:
5532:
5527:
5522:
5517:
5512:
5507:
5502:
5497:
5492:
5487:
5482:
5476:
5474:
5473:Basic concepts
5470:
5469:
5467:
5466:
5461:
5455:
5452:
5451:
5448:Linear algebra
5444:
5443:
5436:
5429:
5421:
5413:
5412:
5373:
5354:(3): 379–384.
5326:
5307:(4): 475–487.
5291:
5280:(4): 365–373.
5264:
5243:
5217:
5203:
5182:
5161:
5147:
5133:
5126:
5106:
5084:
5048:
5022:
4998:
4965:
4942:
4901:
4882:
4841:
4813:
4795:
4777:
4759:
4725:
4690:
4671:
4670:
4668:
4665:
4664:
4663:
4658:
4653:
4648:
4641:
4638:
4621:
4618:
4617:
4616:
4603:
4582:
4578:
4574:
4571:
4568:
4564:
4560:
4556:
4552:
4548:
4520:
4509:
4508:
4497:
4493:
4480:
4475:
4471:
4466:
4462:
4459:
4456:
4452:
4448:
4444:
4440:
4436:
4425:
4414:
4410:
4406:
4403:
4400:
4396:
4392:
4388:
4384:
4381:
4378:
4374:
4361:
4358:
4346:
4342:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4291:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4241:
4238:
4236:
4233:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4197:
4194:
4190:
4186:
4182:
4179:
4176:
4172:
4146:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4095:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4048:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4001:
3998:
3994:
3990:
3981:
3976:
3971:
3967:
3962:
3957:
3953:
3948:
3943:
3942:
3939:
3935:
3931:
3926:
3921:
3915:
3912:
3910:
3907:
3905:
3902:
3901:
3898:
3895:
3893:
3890:
3888:
3885:
3884:
3881:
3878:
3876:
3873:
3871:
3868:
3867:
3865:
3860:
3856:
3843:
3840:
3828:
3824:
3815:
3810:
3805:
3801:
3797:
3795:
3792:
3788:
3783:
3778:
3774:
3770:
3766:
3761:
3756:
3752:
3748:
3747:
3744:
3740:
3736:
3732:
3729:
3726:
3722:
3700:
3679:
3674:
3670:
3666:
3663:
3659:
3641:> 1) with
3625:
3599:
3566:
3563:
3525:
3522:
3398:np.power(a, b)
3266:
3263:
3243:
3240:
3236:
3230:
3227:
3223:
3217:
3214:
3212:
3208:
3205:
3201:
3197:
3196:
3192:
3188:
3184:
3180:
3177:
3175:
3172:
3168:
3167:
3129:
3126:
3119:
3116:
3112:
3106:
3103:
3101:
3097:
3094:
3090:
3086:
3085:
3080:
3077:
3074:
3069:
3064:
3061:
3059:
3056:
3052:
3051:
3014:
3011:
3003:
3000:
2996:
2990:
2987:
2985:
2981:
2978:
2974:
2970:
2969:
2962:
2959:
2954:
2949:
2944:
2941:
2939:
2936:
2932:
2931:
2903:
2896:
2893:
2889:
2883:
2880:
2878:
2874:
2871:
2867:
2863:
2862:
2857:
2854:
2849:
2844:
2841:
2839:
2836:
2832:
2831:
2815:Hadamard power
2800:
2797:
2785:
2782:
2778:
2774:
2771:
2768:
2764:
2760:
2757:
2754:
2751:
2747:
2743:
2739:
2735:
2732:
2697:Main article:
2694:
2691:
2674:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2579: denotes
2568:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2457:
2437:
2417:
2397:
2373:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2273:
2270:
2269:
2268:
2252:
2232:
2211:
2190:
2187:
2184:
2179:
2174:
2168:
2164:
2161:
2158:
2154:
2150:
2147:
2144:
2122:
2110:
2098:
2091:
2086:
2082:
2078:
2071:
2066:
2062:
2058:
2054:
2050:
2024:
2002:
1990:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1916:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1855:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1831:
1808:
1781:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1741:
1738:
1735:
1732:
1729:
1724:
1720:
1714:
1709:
1706:
1703:
1699:
1695:
1692:
1689:
1686:
1683:
1680:
1675:
1671:
1665:
1660:
1657:
1654:
1650:
1622:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1549:
1538:
1522:
1519:
1516:
1513:
1510:
1489:
1482:
1477:
1470:
1465:
1461:
1457:
1453:
1450:
1447:
1443:
1439:
1436:
1433:
1430:
1427:
1405:
1399:
1393:
1388:
1382:
1376:
1371:
1368:
1365:
1361:
1355:
1350:
1344:
1339:
1318:
1313:
1310:
1305:
1298:
1293:
1289:
1285:
1280:
1275:
1272:
1267:
1263:
1257:
1252:
1247:
1242:
1237:
1234:
1230:
1226:
1223:
1220:
1217:
1212:
1208:
1172:
1167:
1163:
1160:
1157:
1153:
1146:
1140:
1135:
1131:
1124:
1119:
1115:
1111:
1107:
1104:
1054:
1050:
1043:
1037:
1029:
1023:
1018:
1013:
1007:
1001:
995:
991:
988:
985:
981:
977:
974:
971:
968:
965:
960:
955:
938:
927:
908:
877:
862:
859:
856:
853:
850:
847:
844:
842:
840:
837:
834:
831:
830:
827:
823:
819:
816:
813:
809:
805:
802:
798:
794:
791:
787:
783:
780:
777:
774:
772:
770:
767:
763:
759:
756:
752:
748:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
715:
713:
710:
707:
704:
701:
698:
695:
692:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
659:
657:
654:
651:
648:
645:
642:
639:
636:
635:
632:
629:
626:
623:
620:
617:
615:
613:
610:
607:
604:
603:
562:
559:
558:
557:
544:
538:
535:
532:
530:
527:
525:
522:
521:
518:
515:
513:
510:
508:
505:
504:
502:
497:
492:
486:
483:
480:
477:
474:
472:
469:
466:
463:
461:
458:
455:
452:
451:
448:
445:
442:
439:
437:
434:
431:
428:
426:
423:
420:
417:
416:
414:
409:
404:
398:
395:
393:
390:
388:
385:
384:
381:
378:
376:
373:
371:
368:
367:
365:
360:
355:
349:
346:
343:
341:
338:
336:
333:
332:
329:
326:
324:
321:
319:
316:
315:
313:
254:
253:
242:
237:
234:
230:
226:
223:
218:
215:
211:
207:
204:
201:
196:
193:
189:
185:
182:
179:
176:
153:
150:
147:
127:
124:
121:
87:
84:
57:matrix product
15:
9:
6:
4:
3:
2:
5886:
5875:
5872:
5870:
5869:Matrix theory
5867:
5866:
5864:
5849:
5841:
5840:
5837:
5831:
5828:
5826:
5825:Sparse matrix
5823:
5821:
5818:
5816:
5813:
5811:
5808:
5807:
5805:
5803:
5799:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5764:
5762:
5760:constructions
5759:
5755:
5749:
5748:Outermorphism
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5708:Cross product
5706:
5704:
5701:
5700:
5698:
5696:
5692:
5686:
5683:
5681:
5678:
5676:
5675:Outer product
5673:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5655:Orthogonality
5653:
5652:
5650:
5648:
5644:
5638:
5635:
5633:
5632:Cramer's rule
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5602:Decomposition
5600:
5598:
5595:
5594:
5592:
5590:
5586:
5581:
5571:
5568:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5548:
5546:
5543:
5541:
5538:
5536:
5533:
5531:
5528:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5477:
5475:
5471:
5465:
5462:
5460:
5457:
5456:
5453:
5449:
5442:
5437:
5435:
5430:
5428:
5423:
5422:
5419:
5408:
5402:
5393:
5388:
5384:
5377:
5369:
5365:
5361:
5357:
5353:
5349:
5342:
5335:
5333:
5331:
5322:
5318:
5314:
5310:
5306:
5302:
5295:
5287:
5283:
5279:
5275:
5268:
5259:
5254:
5247:
5231:
5227:
5221:
5213:
5207:
5192:
5186:
5171:
5165:
5157:
5151:
5143:
5137:
5129:
5127:9781118618363
5123:
5119:
5118:
5110:
5102:
5101:MIT Media Lab
5095:
5088:
5079:
5074:
5070:
5066:
5062:
5055:
5053:
5044:
5040:
5033:
5026:
5008:
5002:
4993:
4988:
4984:
4980:
4976:
4969:
4961:
4957:
4953:
4946:
4938:
4934:
4929:
4924:
4920:
4916:
4912:
4905:
4898:(1): 160–177.
4897:
4893:
4886:
4878:
4873:
4869:
4865:
4861:
4857:
4850:
4848:
4846:
4827:
4820:
4818:
4809:
4805:
4799:
4791:
4787:
4781:
4773:
4769:
4763:
4748:
4741:
4734:
4732:
4730:
4721:
4717:
4713:
4709:
4706:(1): 343–44.
4705:
4701:
4694:
4686:
4679:
4677:
4672:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4643:
4637:
4635:
4631:
4627:
4580:
4576:
4569:
4558:
4554:
4550:
4538:
4537:
4536:
4535:of matrices,
4534:
4518:
4495:
4491:
4473:
4469:
4464:
4457:
4450:
4446:
4442:
4438:
4434:
4426:
4412:
4408:
4401:
4394:
4390:
4386:
4379:
4372:
4364:
4363:
4357:
4344:
4340:
4333:
4328:
4323:
4318:
4313:
4308:
4303:
4298:
4293:
4286:
4281:
4276:
4271:
4266:
4261:
4256:
4251:
4246:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4192:
4188:
4184:
4177:
4170:
4161:
4158:
4144:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4055:
4050:
4043:
4038:
4033:
4028:
4023:
4018:
4013:
4008:
4003:
3996:
3992:
3988:
3979:
3974:
3965:
3960:
3951:
3946:
3937:
3933:
3929:
3924:
3919:
3913:
3908:
3903:
3896:
3891:
3886:
3879:
3874:
3869:
3863:
3858:
3854:
3839:
3826:
3822:
3813:
3808:
3803:
3799:
3793:
3786:
3781:
3776:
3772:
3764:
3759:
3754:
3750:
3742:
3738:
3734:
3727:
3720:
3712:of the form:
3698:
3672:
3668:
3661:
3657:
3648:
3644:
3640:
3623:
3614:
3597:
3588:
3584:
3580:
3571:
3562:
3559:
3557:
3553:
3549:
3544:
3542:
3537:
3535:
3531:
3521:
3515:
3483:
3479:
3475:
3471:
3467:
3463:
3458:
3457:
3453:
3449:
3444:
3430:
3427:), while the
3414:
3409:
3403:
3379:
3366:a.multiply(b)
3359:
3355:
3351:
3342:
3329:
3314:
3310:
3305:
3279:
3274:
3272:
3262:
3241:
3238:
3234:
3228:
3225:
3221:
3215:
3213:
3206:
3203:
3199:
3190:
3186:
3182:
3178:
3176:
3170:
3157:
3155:
3146:
3127:
3124:
3117:
3114:
3110:
3104:
3102:
3095:
3092:
3088:
3078:
3075:
3072:
3067:
3062:
3060:
3054:
3041:
3032:
3012:
3009:
3001:
2998:
2994:
2988:
2986:
2979:
2976:
2972:
2960:
2957:
2952:
2947:
2942:
2940:
2934:
2920:
2901:
2894:
2891:
2887:
2881:
2879:
2872:
2869:
2865:
2855:
2852:
2847:
2842:
2840:
2834:
2820:
2818:
2810:
2807:Hadamard root
2796:
2783:
2776:
2762:
2752:
2745:
2741:
2737:
2722:
2710:
2706:
2700:
2690:
2688:
2672:
2652:
2646:
2643:
2637:
2631:
2628:
2622:
2616:
2613:
2610:
2601:
2598:
2595:
2584:
2582:
2566:
2546:
2540:
2537:
2534:
2528:
2522:
2519:
2516:
2510:
2504:
2501:
2498:
2492:
2486:
2483:
2480:
2469:
2455:
2435:
2415:
2395:
2387:
2371:
2351:
2345:
2342:
2339:
2333:
2327:
2324:
2321:
2315:
2309:
2306:
2303:
2297:
2291:
2288:
2285:
2266:
2250:
2230:
2188:
2185:
2177:
2159:
2145:
2142:
2134:
2120:
2111:
2084:
2080:
2064:
2060:
2052:
2039:
1991:
1973:
1967:
1964:
1961:
1955:
1952:
1949:
1943:
1940:
1934:
1928:
1925:
1919:
1917:
1906:
1903:
1897:
1891:
1888:
1882:
1879:
1876:
1870:
1867:
1864:
1858:
1856:
1851:
1845:
1842:
1839:
1833:
1821:
1809:
1805:
1800:
1795:
1788:
1784:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
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594:is a scalar:
593:
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45:Schur product
42:
38:
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5758:Vector space
5664:
5490:Vector space
5401:cite journal
5382:
5376:
5351:
5347:
5304:
5300:
5294:
5277:
5273:
5267:
5246:
5234:. Retrieved
5229:
5220:
5206:
5194:. Retrieved
5185:
5173:. Retrieved
5164:
5150:
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5100:
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5014:. Retrieved
5001:
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4859:
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4832:. Retrieved
4807:
4798:
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4780:
4771:
4762:
4752:September 6,
4750:. Retrieved
4746:
4703:
4699:
4693:
4684:
4623:
4620:Applications
4615:is a vector.
4531:denotes the
4510:
4162:
4159:
3845:
3646:
3642:
3638:
3612:
3586:
3582:
3576:
3560:
3545:
3538:
3527:
3524:Applications
3459:
3455:
3445:
3417:cwiseProduct
3411:In C++, the
3410:
3352:
3324:
3312:
3306:
3275:
3268:
3158:
3149:
3147:
3035:
3033:
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1071:denotes the
1067:
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924:
918:
912:
910:For vectors
896:
892:
885:
881:
591:
587:
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579:
576:distributive
298:
292:
288:
282:
278:
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76:distributive
69:
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5874:Issai Schur
5738:Multivector
5703:Determinant
5660:Dot product
5505:Linear span
5045:(3): 50–53.
4985:: 313–320.
4862:: 217–240,
3482:Mathematica
3374:a.matmul(b)
3286:times(a, b)
2721:determinant
2709:Issai Schur
2388:, assuming
1797:th largest
1329:Similarly,
1095:, that is,
572:associative
568:commutative
138:(sometimes
80:commutative
72:associative
65:Issai Schur
29:mathematics
5863:Categories
5772:Direct sum
5607:Invertible
5510:Linear map
5392:1609.09106
5385:: Page 6.
5274:Statistics
5196:31 January
5175:31 January
5016:2019-12-18
4921:: 104849.
4667:References
3579:V. Slyusar
3514:matrixcalc
3437:a % b
1799:eigenvalue
561:Properties
86:Definition
5802:Numerical
5565:Transpose
5368:119661450
5321:121385730
5258:1402.1128
5236:24 August
5071:: 35–43.
5007:"Project"
4937:239598156
4834:2 January
4720:121027182
4570:∘
4551:∙
4519:∙
4474:⊗
4458:∘
4439:∙
4402:∘
4380:∘
4178:∘
3804:∘
3794:⋯
3777:∘
3755:∘
3728:∘
3429:Armadillo
3400:, or the
3187:⊘
3125:−
3076:−
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2953:∘
2853:∘
2753:≥
2742:⊙
2673:∗
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2614:∗
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2567:∙
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2372:⊗
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2307:⊗
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2289:⊗
2186:⊙
2146:
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1956:⊙
1935:⊙
1898:⊙
1877:⊙
1843:⊙
1756:…
1719:λ
1698:∏
1694:≥
1685:⊙
1670:λ
1649:∏
1601:
1586:
1580:≤
1571:⊙
1562:
1542:submatrix
1512:
1452:
1432:⊙
1404:∗
1364:⊙
1354:∗
1222:∘
1207:∑
1159:⊙
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650:⊙
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625:⊙
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482:×
476:−
468:×
457:×
444:×
433:×
422:×
359:∘
345:−
181:⊙
149:∘
123:⊙
5848:Category
5787:Subspace
5782:Quotient
5733:Bivector
5647:Bilinear
5589:Matrices
5464:Glossary
4640:See also
3649:blocks (
3452:HP Prime
3406:a.pow(b)
2922:and for
2133:operator
276:, where
53:matrices
5459:Outline
3842:Example
3589:matrix
3462:Fortran
3423:class (
3404:method
3042:reads:
2263:is the
1822:, then
1791:is the
1544:of the
47:) is a
5743:Tensor
5555:Kernel
5485:Vector
5480:Scalar
5366:
5319:
5124:
4935:
4718:
4626:tensor
4593:where
4511:where
3494:matmul
3450:, and
3421:Matrix
3402:Pandas
3354:Python
3321:a .^ b
3317:a .* b
3294:a ./ b
3290:a .^ b
3282:a .* b
3278:MATLAB
2665:where
2559:where
2364:where
2201:where
1777:where
1501:where
1065:where
586:, and
31:, the
5612:Minor
5597:Block
5535:Basis
5387:arXiv
5364:S2CID
5344:(PDF)
5317:S2CID
5253:arXiv
5097:(PDF)
5035:(PDF)
5010:(PDF)
4933:S2CID
4829:(PDF)
4743:(PDF)
4716:S2CID
4160:then
3556:LSTMs
3506:+/ .*
3448:GAUSS
3441:a * b
3413:Eigen
3382:array
3378:SymPy
3358:NumPy
3348:f.(x)
3309:Julia
3300:and
2448:with
1083:trace
907:zero.
5767:Dual
5622:Rank
5407:link
5238:2013
5198:2024
5177:2024
5122:ISBN
4836:2012
4754:2020
3611:and
3552:GRUs
3534:JPEG
3508:and
3476:and
3364:or
3335:and
3292:and
3034:The
2822:For
2811:and
2715:and
2428:and
2243:and
2143:diag
2121:diag
2014:and
1818:are
1814:and
1636:are
1630:and
1598:rank
1583:rank
1559:rank
1509:diag
1449:diag
1188:and
933:and
916:and
574:and
266:and
96:and
74:and
5356:doi
5309:doi
5282:doi
5073:doi
5069:288
4987:doi
4983:515
4956:doi
4923:doi
4919:188
4872:hdl
4864:doi
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4282:150
4272:240
4267:125
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3554:or
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3502:+.×
3498:%*%
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2756:det
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584:B
580:A
543:]
524:0
517:4
512:3
507:6
501:[
496:=
491:]
485:5
479:2
471:9
465:8
460:7
454:0
447:4
441:1
436:1
430:3
425:3
419:2
413:[
408:=
403:]
397:5
392:9
387:7
380:4
375:1
370:3
364:[
354:]
348:2
340:8
335:0
328:1
323:3
318:2
312:[
293:q
289:n
283:p
279:m
273:q
269:p
263:n
259:m
241:.
236:j
233:i
229:)
225:B
222:(
217:j
214:i
210:)
206:A
203:(
200:=
195:j
192:i
188:)
184:B
178:A
175:(
152:B
146:A
126:B
120:A
109:n
105:m
99:B
93:A
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