2591:
678:
1943:
1641:
2400:
2071:
1957:
from one vector space to another forms a vector space itself. One can therefore take the tensor product of linear transformations. Such a tensor product can be naturally interpreted as a linear transformation on a tensor product.
3085:
1361:
1758:
2271:
409:
318:
756:
2473:
1476:
2157:
487:
1200:
914:
186:
498:
1011:
1769:
2742:
2781:
2638:
1130:
2909:
2851:
1523:
2668:
1156:
2783:. So, the tensors of the linear functionals are bilinear functionals. This gives us a new way to look at the space of bilinear functionals, as a tensor product itself.
2698:
1082:
1049:
2282:
1223:
2465:
2445:
3190:
However, these kinds of notation are not universally present in all array languages. Some languages may require explicit treatment of indices (for example,
2917:
683:
Note that the symbol ⊗ is overloaded: it is used to denote the tensor product of two vectors as well as the tensor product of two vector spaces.
3157:
for( int i = 0; i < i_dim; i++) for( int j = 0; j < j_dim; j++) for( int k = 0; k < k_dim; k++) result = a*b;
1293:
1235:
1010:
1660:
2179:
1967:
324:
233:
2586:{\displaystyle U\otimes V={\begin{bmatrix}u_{11}V&u_{12}V&\cdots \\u_{21}V&u_{22}V\\\vdots &&\ddots \end{bmatrix}}}
704:
2427:
imposed upon it, in which each element of the first matrix is replaced by the second matrix, scaled by that element. For matrices
2169:
There are several special cases of this embedding which can be obtained by taking one or more of the spaces to be the ground field
3236:
1410:
2082:
673:{\displaystyle (v_{1}+v_{2})\otimes (w_{1}+w_{2})=v_{1}\otimes w_{1}+v_{1}\otimes w_{2}+v_{2}\otimes w_{1}+v_{2}\otimes w_{2}.}
415:
1161:
3246:
872:
144:
1938:{\displaystyle (X\otimes Y)^{i_{1}i_{2}\cdots i_{k+\ell }}=X^{i_{1}i_{2}\cdots i_{k}}Y^{i_{k+1}i_{k+2}\cdots i_{k+\ell }}.}
3226:
2700:. In other words, every bilinear functional is a functional on the tensor product, and vice versa. There is a natural
3241:
3161:
1158:. The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance
2707:
1763:
In terms of these coordinates the tensor product is just given by ordinary multiplication of components. That is,
3221:
3266:
3231:
3212:
select a.i as i, a.j as j, b.k as k, a.value * b.value as value from a outer join b
3165:
2671:
3262:
3256:
2163:
2747:
2604:
3173:
3131:
2166:
is zero. If all spaces in question are finite-dimensional, this map is a actually a linear isomorphism.
1954:
1095:
112:
2856:
2798:
1651:
1636:{\displaystyle e_{i_{1}i_{2}\cdots i_{k}}=e_{i_{1}}\otimes e_{i_{2}}\otimes \cdots \otimes e_{i_{k}}}
1255:
840:
66:
21:
1004:
200:
2647:
1135:
2677:
1054:
1021:
115:
in each argument. That is, fixing either the first or second argument results in a linear map.
2412:
2395:{\displaystyle V^{*}\otimes W^{*}\to (V\otimes W)^{*}\qquad (f\otimes g)(v\otimes w)=f(v)g(w)}
97:
1391:
779:
2424:
823:. The tensor product (of tensors) is the natural multiplication operator in this algebra.
1205:
2450:
2430:
815:
38:
3080:{\displaystyle (f\otimes g)(x_{1},...x_{k+m})=f(x_{1},...x_{k})g(x_{k+1},...x_{k+m})}
49:, denoted by the symbol ⊗, refers to a number of closely related operations in
1485:
tensor and a rank ℓ tensor their tensor product can be interpreted as a rank
27:
1272:
1251:
924:
766:
227:, subject only to the relations necessary for bilinearity. That is, one must have:
82:
17:
3107:
is the rank of the tensor. The tensor product is an operation which takes a rank-
2674:
on that space) corresponds naturally to the space of all bilinear functionals on
3181:
2792:
2701:
1404:
Associativity of the tensor product means that there is a canonical isomorphism
58:
50:
2641:
946:
806:
53:. The unifying concept is that the tensor product represents the most general
3127:
65:, tensor spaces, and tensor products forms a branch of mathematics called
78:
54:
1356:{\displaystyle T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.}
1262:
919:
The universality of this map means the following: for any vector space
77:
The tensor product can be thought of as the “most general”
1753:{\displaystyle X=X^{i_{1}i_{2}\cdots i_{k}}e_{i_{1}i_{2}\cdots i_{k}}}
3191:
1386:
775:
62:
2266:{\displaystyle V^{*}\otimes W\to L(V,W)\qquad (f\otimes w)(v)=f(v)w}
2066:{\displaystyle L(V,V')\otimes L(W,W')\to L(V\otimes W,V'\otimes W')}
404:{\displaystyle v\otimes (w_{1}+w_{2})=v\otimes w_{1}+v\otimes w_{2}}
313:{\displaystyle (v_{1}+v_{2})\otimes w=v_{1}\otimes w+v_{2}\otimes w}
3100:
1000:
3095:
Tensors in computer programming are almost always represented as
1377:(as a one-dimensional vector space over itself). The elements of
751:{\displaystyle V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.}
118:
The tensor product on vectors is defined to be a bilinear map on
3198:
3197:
Another interesting example of the tensor product comes from
2423:, a term used to make clear that the result has a particular
1009:
1471:{\displaystyle T^{k}V\otimes T^{\ell }V\cong T^{k+\ell }V.}
2152:{\displaystyle (f\otimes g)(v\otimes w)=f(v)\otimes g(w).}
2162:
This map is a linear embedding, which is to say that its
482:{\displaystyle k(v\otimes w)=(kv)\otimes w=v\otimes (kw)}
1195:{\displaystyle \mathbb {R} ^{m}\otimes \mathbb {R} ^{n}}
909:{\displaystyle \otimes \colon V\times W\to V\otimes W.}
181:{\displaystyle \otimes \colon V\times W\to V\otimes W.}
2494:
2920:
2859:
2801:
2750:
2710:
2680:
2650:
2607:
2476:
2453:
2433:
2285:
2182:
2085:
1970:
1772:
1663:
1526:
1413:
1296:
1208:
1164:
1138:
1098:
1057:
1024:
875:
707:
501:
418:
327:
236:
147:
2419:With matrices this operation is usually called the
1961:Specifically, there exists is a natural linear map
3079:
2911:their tensor product is the multilinear function
2903:
2845:
2775:
2736:
2692:
2662:
2632:
2585:
2459:
2439:
2394:
2265:
2151:
2065:
1937:
1752:
1635:
1470:
1355:
1217:
1194:
1150:
1124:
1076:
1043:
908:
750:
672:
481:
403:
312:
180:
3164:may have this pattern built in. For example, in
1014:Universal property of the tensor product space
8:
2737:{\displaystyle V^{\star }\otimes W^{\star }}
1071:
1058:
1038:
1025:
2173:. Specifically, one has natural embeddings
492:Combining the first two properties we have
774:. The elements of such a space are called
3062:
3034:
3015:
2993:
2965:
2943:
2919:
2892:
2870:
2858:
2834:
2812:
2800:
2767:
2749:
2728:
2715:
2709:
2679:
2649:
2624:
2606:
2553:
2538:
2516:
2501:
2489:
2475:
2452:
2432:
2328:
2303:
2290:
2284:
2187:
2181:
2084:
1969:
1918:
1899:
1883:
1878:
1866:
1853:
1843:
1838:
1817:
1804:
1794:
1789:
1771:
1742:
1729:
1719:
1714:
1702:
1689:
1679:
1674:
1662:
1625:
1620:
1599:
1594:
1579:
1574:
1559:
1546:
1536:
1531:
1525:
1450:
1434:
1418:
1412:
1317:
1301:
1295:
1207:
1186:
1182:
1181:
1171:
1167:
1166:
1163:
1137:
1116:
1103:
1097:
1065:
1056:
1032:
1023:
874:
712:
706:
661:
648:
635:
622:
609:
596:
583:
570:
554:
541:
522:
509:
500:
417:
395:
376:
354:
341:
326:
298:
279:
257:
244:
235:
146:
3115:tensor and computes the resulting rank-(
1949:Tensor product of linear transformations
3267:question marks, boxes, or other symbols
3176:the tensor product is the expressed as
1092:respectively, the tensors of the form
96:, a bilinear map is a function on the
3130:for doing this is given below in the
2776:{\displaystyle (V\otimes W)^{\star }}
2633:{\displaystyle (V\otimes W)^{\star }}
1236:tensor product of modules over a ring
690:, one can take the tensor product of
7:
3209:can be given by the following code:
81:product on a pair of vectors. Given
3168:the tensor product is expressed as
3142:is a rank-one tensor, with indices
2787:Tensor product of multilinear maps
1125:{\displaystyle v_{i}\otimes w_{j}}
1003:a unique isomorphism by the above
761:The resulting space is called the
35:
3184:, the tensor product is given by
2904:{\displaystyle g(x_{1},...x_{m})}
2846:{\displaystyle f(x_{1},...x_{k})}
3237:tensor product of Hilbert spaces
2334:
2220:
827:Tensor product of vector spaces
813:one obtains a space called the
789:. The tensor product of a rank
3247:tensor product of line bundles
3074:
3027:
3021:
2986:
2977:
2936:
2933:
2921:
2898:
2863:
2840:
2805:
2764:
2751:
2621:
2608:
2389:
2383:
2377:
2371:
2362:
2350:
2347:
2335:
2325:
2312:
2309:
2257:
2251:
2242:
2236:
2233:
2221:
2217:
2205:
2199:
2143:
2137:
2128:
2122:
2113:
2101:
2098:
2086:
2060:
2026:
2020:
2017:
2000:
1991:
1974:
1786:
1773:
891:
839:be vector spaces over a fixed
560:
534:
528:
502:
476:
467:
449:
440:
434:
422:
360:
334:
263:
237:
163:
1:
3201:. Here the tensor product of
1650:tensor is written (using the
1481:In other words, given a rank
858:is another vector space over
3227:tensor product of R-algebras
2597:Relation with the dual space
199:can be thought of as formal
3162:array programming languages
797:tensor is naturally a rank
686:Given a fixed vector space
3286:
3242:topological tensor product
2663:{\displaystyle V\otimes W}
1151:{\displaystyle V\otimes W}
3222:tensor product of modules
3138:is a rank-two tensor and
2693:{\displaystyle V\times W}
1279:is the tensor product of
1242:Tensor product of tensors
1077:{\displaystyle \{w_{i}\}}
1044:{\displaystyle \{v_{i}\}}
991:The tensor product space
3232:tensor product of fields
809:of all tensor powers of
61:) product. The study of
3132:C programming language
3081:
2905:
2847:
2777:
2738:
2694:
2670:containing all linear
2664:
2634:
2587:
2461:
2441:
2396:
2267:
2153:
2067:
1955:linear transformations
1939:
1754:
1637:
1493:Coordinate description
1472:
1357:
1261:. For any nonnegative
1219:
1196:
1152:
1126:
1078:
1045:
1015:
910:
805:tensor. By taking the
752:
674:
483:
405:
314:
182:
126:to a new vector space
3082:
2906:
2848:
2778:
2739:
2695:
2665:
2635:
2588:
2462:
2442:
2397:
2268:
2154:
2068:
1940:
1755:
1638:
1473:
1358:
1220:
1197:
1153:
1127:
1079:
1046:
1013:
911:
753:
675:
484:
406:
315:
183:
3091:Computer programming
2918:
2857:
2799:
2748:
2708:
2678:
2648:
2605:
2601:Note that the space
2474:
2451:
2431:
2283:
2180:
2083:
1968:
1770:
1661:
1652:summation convention
1524:
1411:
1373:is the ground field
1294:
1206:
1202:will have dimension
1162:
1136:
1096:
1055:
1022:
873:
705:
499:
416:
325:
234:
145:
136:tensor product space
111:which is separately
28:User:Fropuff/Draft 8
3255:This page contains
3170:A ∘.× B
1489:+ ℓ tensor.
201:linear combinations
67:multilinear algebra
3257:special characters
3111:tensor and a rank-
3103:of numbers, where
3077:
2901:
2843:
2773:
2734:
2690:
2660:
2630:
2583:
2577:
2457:
2437:
2392:
2263:
2149:
2063:
1935:
1750:
1633:
1468:
1353:
1218:{\displaystyle mn}
1215:
1192:
1148:
1132:forms a basis for
1122:
1074:
1041:
1016:
1005:universal property
906:
793:tensor and a rank
748:
670:
479:
401:
310:
178:
3263:rendering support
2460:{\displaystyle V}
2440:{\displaystyle U}
2421:Kronecker product
2413:Kronecker product
2406:Kronecker product
98:cartesian product
26:(Redirected from
3277:
3179:
3171:
3154:, respectively.
3086:
3084:
3083:
3078:
3073:
3072:
3045:
3044:
3020:
3019:
2998:
2997:
2976:
2975:
2948:
2947:
2910:
2908:
2907:
2902:
2897:
2896:
2875:
2874:
2852:
2850:
2849:
2844:
2839:
2838:
2817:
2816:
2782:
2780:
2779:
2774:
2772:
2771:
2743:
2741:
2740:
2735:
2733:
2732:
2720:
2719:
2699:
2697:
2696:
2691:
2669:
2667:
2666:
2661:
2639:
2637:
2636:
2631:
2629:
2628:
2592:
2590:
2589:
2584:
2582:
2581:
2570:
2558:
2557:
2543:
2542:
2521:
2520:
2506:
2505:
2466:
2464:
2463:
2458:
2446:
2444:
2443:
2438:
2401:
2399:
2398:
2393:
2333:
2332:
2308:
2307:
2295:
2294:
2272:
2270:
2269:
2264:
2192:
2191:
2158:
2156:
2155:
2150:
2072:
2070:
2069:
2064:
2059:
2048:
2016:
1990:
1944:
1942:
1941:
1936:
1931:
1930:
1929:
1928:
1910:
1909:
1894:
1893:
1873:
1872:
1871:
1870:
1858:
1857:
1848:
1847:
1830:
1829:
1828:
1827:
1809:
1808:
1799:
1798:
1759:
1757:
1756:
1751:
1749:
1748:
1747:
1746:
1734:
1733:
1724:
1723:
1709:
1708:
1707:
1706:
1694:
1693:
1684:
1683:
1642:
1640:
1639:
1634:
1632:
1631:
1630:
1629:
1606:
1605:
1604:
1603:
1586:
1585:
1584:
1583:
1566:
1565:
1564:
1563:
1551:
1550:
1541:
1540:
1477:
1475:
1474:
1469:
1461:
1460:
1439:
1438:
1423:
1422:
1362:
1360:
1359:
1354:
1325:
1324:
1306:
1305:
1224:
1222:
1221:
1216:
1201:
1199:
1198:
1193:
1191:
1190:
1185:
1176:
1175:
1170:
1157:
1155:
1154:
1149:
1131:
1129:
1128:
1123:
1121:
1120:
1108:
1107:
1083:
1081:
1080:
1075:
1070:
1069:
1050:
1048:
1047:
1042:
1037:
1036:
915:
913:
912:
907:
862:together with a
757:
755:
754:
749:
720:
719:
679:
677:
676:
671:
666:
665:
653:
652:
640:
639:
627:
626:
614:
613:
601:
600:
588:
587:
575:
574:
559:
558:
546:
545:
527:
526:
514:
513:
488:
486:
485:
480:
410:
408:
407:
402:
400:
399:
381:
380:
359:
358:
346:
345:
319:
317:
316:
311:
303:
302:
284:
283:
262:
261:
249:
248:
191:The elements of
187:
185:
184:
179:
31:
3285:
3284:
3280:
3279:
3278:
3276:
3275:
3274:
3273:
3272:
3271:
3270:
3261:Without proper
3218:
3213:
3188:
3177:
3169:
3158:
3126:A prototypical
3093:
3058:
3030:
3011:
2989:
2961:
2939:
2916:
2915:
2888:
2866:
2855:
2854:
2830:
2808:
2797:
2796:
2789:
2763:
2746:
2745:
2724:
2711:
2706:
2705:
2676:
2675:
2646:
2645:
2620:
2603:
2602:
2599:
2576:
2575:
2569:
2563:
2562:
2549:
2547:
2534:
2531:
2530:
2525:
2512:
2510:
2497:
2490:
2472:
2471:
2449:
2448:
2429:
2428:
2425:block structure
2408:
2324:
2299:
2286:
2281:
2280:
2183:
2178:
2177:
2081:
2080:
2052:
2041:
2009:
1983:
1966:
1965:
1953:The set of all
1951:
1914:
1895:
1879:
1874:
1862:
1849:
1839:
1834:
1813:
1800:
1790:
1785:
1768:
1767:
1738:
1725:
1715:
1710:
1698:
1685:
1675:
1670:
1659:
1658:
1646:A general rank
1621:
1616:
1595:
1590:
1575:
1570:
1555:
1542:
1532:
1527:
1522:
1521:
1505:
1497:Given a basis {
1495:
1446:
1430:
1414:
1409:
1408:
1313:
1297:
1292:
1291:
1244:
1231:
1229:Generalizations
1204:
1203:
1180:
1165:
1160:
1159:
1134:
1133:
1112:
1099:
1094:
1093:
1061:
1053:
1052:
1028:
1020:
1019:
942:there exists a
871:
870:
829:
708:
703:
702:
657:
644:
631:
618:
605:
592:
579:
566:
550:
537:
518:
505:
497:
496:
414:
413:
391:
372:
350:
337:
323:
322:
294:
275:
253:
240:
232:
231:
143:
142:
75:
42:
33:
32:
25:
24:
12:
11:
5:
3283:
3281:
3265:, you may see
3253:
3252:
3251:
3250:
3249:
3244:
3239:
3234:
3229:
3224:
3217:
3214:
3211:
3186:
3156:
3092:
3089:
3088:
3087:
3076:
3071:
3068:
3065:
3061:
3057:
3054:
3051:
3048:
3043:
3040:
3037:
3033:
3029:
3026:
3023:
3018:
3014:
3010:
3007:
3004:
3001:
2996:
2992:
2988:
2985:
2982:
2979:
2974:
2971:
2968:
2964:
2960:
2957:
2954:
2951:
2946:
2942:
2938:
2935:
2932:
2929:
2926:
2923:
2900:
2895:
2891:
2887:
2884:
2881:
2878:
2873:
2869:
2865:
2862:
2842:
2837:
2833:
2829:
2826:
2823:
2820:
2815:
2811:
2807:
2804:
2788:
2785:
2770:
2766:
2762:
2759:
2756:
2753:
2731:
2727:
2723:
2718:
2714:
2689:
2686:
2683:
2659:
2656:
2653:
2627:
2623:
2619:
2616:
2613:
2610:
2598:
2595:
2594:
2593:
2580:
2574:
2571:
2568:
2565:
2564:
2561:
2556:
2552:
2548:
2546:
2541:
2537:
2533:
2532:
2529:
2526:
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2411:Main article:
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2100:
2097:
2094:
2091:
2088:
2076:determined by
2074:
2073:
2062:
2058:
2055:
2051:
2047:
2044:
2040:
2037:
2034:
2031:
2028:
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2019:
2015:
2012:
2008:
2005:
2002:
1999:
1996:
1993:
1989:
1986:
1982:
1979:
1976:
1973:
1950:
1947:
1946:
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1934:
1927:
1924:
1921:
1917:
1913:
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1569:
1562:
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1549:
1545:
1539:
1535:
1530:
1510:, a basis for
1501:
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1479:
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1456:
1453:
1449:
1445:
1442:
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1433:
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1366:By convention
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1111:
1106:
1102:
1073:
1068:
1064:
1060:
1040:
1035:
1031:
1027:
999:is determined
917:
916:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
848:tensor product
828:
825:
816:tensor algebra
759:
758:
747:
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74:
71:
47:tensor product
43:
41:
39:Tensor product
36:
34:
15:
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10:
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3099:-dimensional
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2013:
2010:
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1997:
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869:
868:
867:
866:bilinear map
865:
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834:
826:
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431:
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373:
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331:
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321:
307:
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288:
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280:
276:
272:
269:
266:
258:
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222:
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172:
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148:
141:
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139:
137:
133:
129:
125:
121:
116:
114:
110:
106:
102:
99:
95:
91:
87:
84:
83:vector spaces
80:
72:
70:
68:
64:
60:
56:
52:
48:
40:
37:
29:
23:
19:
3254:
3206:
3202:
3196:
3189:
3159:
3151:
3147:
3143:
3139:
3135:
3125:
3120:
3116:
3112:
3108:
3104:
3096:
3094:
2790:
2600:
2420:
2418:
2410:
2409:
2275:
2170:
2168:
2161:
2075:
1960:
1952:
1762:
1647:
1645:
1517:is given by
1514:
1511:
1507:
1502:
1498:
1496:
1486:
1482:
1480:
1403:
1398:
1394:
1385:
1381:
1378:
1374:
1370:
1367:
1365:
1284:
1283:with itself
1280:
1276:
1273:tensor power
1269:
1265:
1258:
1252:vector space
1247:
1245:
1233:
1232:
1089:
1085:
1018:Given bases
1017:
996:
992:
990:
985:
981:
977:
973:
969:
965:
961:
957:
953:
949:
943:
939:
935:
931:
927:
925:bilinear map
920:
918:
863:
859:
855:
851:
847:
843:
836:
832:
830:
820:
814:
810:
802:
798:
794:
790:
786:
782:
771:
767:tensor power
765:
762:
760:
695:
694:with itself
691:
687:
685:
682:
491:
224:
220:
216:
212:
208:
204:
196:
192:
190:
135:
131:
127:
123:
119:
117:
108:
104:
100:
93:
89:
85:
76:
46:
44:
18:User:Fropuff
3182:Mathematica
3172:, while in
2793:multilinear
2702:isomorphism
2672:functionals
1384:are called
203:of symbols
134:called the
59:multilinear
51:mathematics
3123:) tensor.
2642:dual space
964:such that
947:linear map
923:, and any
807:direct sum
3128:algorithm
2928:⊗
2769:⋆
2758:⊗
2730:⋆
2722:⊗
2717:⋆
2704:between
2685:×
2655:⊗
2626:⋆
2615:⊗
2573:⋱
2567:⋮
2528:⋯
2481:⊗
2467:this is:
2357:⊗
2342:⊗
2330:∗
2319:⊗
2310:→
2305:∗
2297:⊗
2292:∗
2228:⊗
2200:→
2194:⊗
2189:∗
2132:⊗
2108:⊗
2093:⊗
2050:⊗
2033:⊗
2021:→
1995:⊗
1926:ℓ
1912:⋯
1860:⋯
1825:ℓ
1811:⋯
1780:⊗
1736:⋯
1696:⋯
1614:⊗
1611:⋯
1608:⊗
1588:⊗
1553:⋯
1458:ℓ
1444:≅
1436:ℓ
1428:⊗
1345:⊗
1342:⋯
1339:⊗
1333:⊗
1319:⊗
1178:⊗
1143:⊗
1110:⊗
898:⊗
892:→
886:×
880::
877:⊗
864:universal
740:⊗
737:⋯
734:⊗
728:⊗
714:⊗
655:⊗
629:⊗
603:⊗
577:⊗
532:⊗
465:⊗
453:⊗
429:⊗
389:⊗
370:⊗
332:⊗
305:⊗
286:⊗
267:⊗
170:⊗
164:→
158:×
152::
149:⊗
3216:See also
2057:′
2046:′
2014:′
1988:′
960:→
938:→
223:∈
215:∈
79:bilinear
73:Overview
55:bilinear
20: |
3187:Outer
3134:. Here
1387:tensors
1287:times:
1263:integer
1254:over a
995:⊗
972:⊗
956:⊗
776:tensors
698:times:
207:⊗
195:⊗
130:⊗
103:×
63:tensors
3192:Matlab
3178:a */ b
3150:, and
3101:arrays
2791:Given
2164:kernel
1654:) as:
1506:} for
1268:, the
944:unique
934:×
846:. The
113:linear
92:, and
22:Drafts
3180:. In
3160:Many
2795:maps
2640:(the
1256:field
1250:be a
1001:up to
841:field
16:<
3205:and
2853:and
2744:and
2447:and
2276:and
1392:rank
1246:Let
1234:See
1088:and
1084:for
1051:and
976:) =
854:and
835:and
831:Let
780:rank
219:and
211:for
122:and
57:(or
45:The
3199:SQL
3194:).
3166:APL
2644:of
1397:on
1390:of
1275:of
988:).
850:of
819:of
785:on
778:of
770:of
107:to
3146:,
2555:22
2540:21
2518:12
2503:11
1401:.
1225:.
1007:.
984:,
952::
930::
138::
88:,
69:.
3269:.
3259:.
3207:b
3203:a
3174:J
3152:k
3148:j
3144:i
3140:b
3136:a
3121:m
3119:+
3117:n
3113:m
3109:n
3105:n
3097:n
3075:)
3070:m
3067:+
3064:k
3060:x
3056:.
3053:.
3050:.
3047:,
3042:1
3039:+
3036:k
3032:x
3028:(
3025:g
3022:)
3017:k
3013:x
3009:.
3006:.
3003:.
3000:,
2995:1
2991:x
2987:(
2984:f
2981:=
2978:)
2973:m
2970:+
2967:k
2963:x
2959:.
2956:.
2953:.
2950:,
2945:1
2941:x
2937:(
2934:)
2931:g
2925:f
2922:(
2899:)
2894:m
2890:x
2886:.
2883:.
2880:.
2877:,
2872:1
2868:x
2864:(
2861:g
2841:)
2836:k
2832:x
2828:.
2825:.
2822:.
2819:,
2814:1
2810:x
2806:(
2803:f
2765:)
2761:W
2755:V
2752:(
2726:W
2713:V
2688:W
2682:V
2658:W
2652:V
2622:)
2618:W
2612:V
2609:(
2579:]
2560:V
2551:u
2545:V
2536:u
2523:V
2514:u
2508:V
2499:u
2492:[
2487:=
2484:V
2478:U
2455:V
2435:U
2415:.
2390:)
2387:w
2384:(
2381:g
2378:)
2375:v
2372:(
2369:f
2366:=
2363:)
2360:w
2354:v
2351:(
2348:)
2345:g
2339:f
2336:(
2326:)
2322:W
2316:V
2313:(
2301:W
2288:V
2261:w
2258:)
2255:v
2252:(
2249:f
2246:=
2243:)
2240:v
2237:(
2234:)
2231:w
2225:f
2222:(
2218:)
2215:W
2212:,
2209:V
2206:(
2203:L
2197:W
2185:V
2171:K
2147:.
2144:)
2141:w
2138:(
2135:g
2129:)
2126:v
2123:(
2120:f
2117:=
2114:)
2111:w
2105:v
2102:(
2099:)
2096:g
2090:f
2087:(
2061:)
2054:W
2043:V
2039:,
2036:W
2030:V
2027:(
2024:L
2018:)
2011:W
2007:,
2004:W
2001:(
1998:L
1992:)
1985:V
1981:,
1978:V
1975:(
1972:L
1933:.
1923:+
1920:k
1916:i
1907:2
1904:+
1901:k
1897:i
1891:1
1888:+
1885:k
1881:i
1876:Y
1868:k
1864:i
1855:2
1851:i
1845:1
1841:i
1836:X
1832:=
1822:+
1819:k
1815:i
1806:2
1802:i
1796:1
1792:i
1787:)
1783:Y
1777:X
1774:(
1744:k
1740:i
1731:2
1727:i
1721:1
1717:i
1712:e
1704:k
1700:i
1691:2
1687:i
1681:1
1677:i
1672:X
1668:=
1665:X
1648:k
1627:k
1623:i
1618:e
1601:2
1597:i
1592:e
1581:1
1577:i
1572:e
1568:=
1561:k
1557:i
1548:2
1544:i
1538:1
1534:i
1529:e
1515:V
1512:T
1508:V
1503:i
1499:e
1487:k
1483:k
1466:.
1463:V
1455:+
1452:k
1448:T
1441:V
1432:T
1425:V
1420:k
1416:T
1399:V
1395:k
1382:V
1379:T
1375:K
1371:V
1368:T
1351:.
1348:V
1336:V
1330:V
1327:=
1322:k
1315:V
1311:=
1308:V
1303:k
1299:T
1285:k
1281:V
1277:V
1270:k
1266:k
1259:K
1248:V
1213:n
1210:m
1188:n
1183:R
1173:m
1168:R
1146:W
1140:V
1118:j
1114:w
1105:i
1101:v
1090:W
1086:V
1072:}
1067:i
1063:w
1059:{
1039:}
1034:i
1030:v
1026:{
997:W
993:V
986:y
982:x
980:(
978:B
974:w
970:v
968:(
966:L
962:X
958:W
954:V
950:L
940:X
936:W
932:V
928:B
921:X
904:.
901:W
895:V
889:W
883:V
860:K
856:W
852:V
844:K
837:W
833:V
821:V
811:V
803:n
801:+
799:m
795:n
791:m
787:V
783:k
772:V
763:k
746:.
743:V
731:V
725:V
722:=
717:k
710:V
696:k
692:V
688:V
668:.
663:2
659:w
650:2
646:v
642:+
637:1
633:w
624:2
620:v
616:+
611:2
607:w
598:1
594:v
590:+
585:1
581:w
572:1
568:v
564:=
561:)
556:2
552:w
548:+
543:1
539:w
535:(
529:)
524:2
520:v
516:+
511:1
507:v
503:(
477:)
474:w
471:k
468:(
462:v
459:=
456:w
450:)
447:v
444:k
441:(
438:=
435:)
432:w
426:v
423:(
420:k
397:2
393:w
386:v
383:+
378:1
374:w
367:v
364:=
361:)
356:2
352:w
348:+
343:1
339:w
335:(
329:v
308:w
300:2
296:v
292:+
289:w
281:1
277:v
273:=
270:w
264:)
259:2
255:v
251:+
246:1
242:v
238:(
225:W
221:w
217:V
213:v
209:w
205:v
197:W
193:V
176:.
173:W
167:V
161:W
155:V
132:W
128:V
124:W
120:V
109:X
105:W
101:V
94:X
90:W
86:V
30:)
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