1225:
961:
1751:
1220:{\displaystyle {\begin{aligned}\operatorname {vec} (ABC)&=(I_{n}\otimes AB)\operatorname {vec} (C)=(C^{\mathrm {T} }B^{\mathrm {T} }\otimes I_{k})\operatorname {vec} (A)\\\operatorname {vec} (AB)&=(I_{m}\otimes A)\operatorname {vec} (B)=(B^{\mathrm {T} }\otimes I_{k})\operatorname {vec} (A)\end{aligned}}}
1616:
915:
2017:
1892:
1458:
429:
1338:
2641:
673:
566:
2439:
2371:
491:
2543:
and its applications in establishing e.g., moments of random vectors and matrices, asymptotics, as well as
Jacobian and Hessian matrices. It is also used in local sensitivity and statistical diagnostics.
1932:
2310:
266:
1810:
802:
1378:
1746:{\displaystyle \mathbf {B} _{i}={\begin{bmatrix}\mathbf {0} \\\vdots \\\mathbf {0} \\\mathbf {I} _{m}\\\mathbf {0} \\\vdots \\\mathbf {0} \end{bmatrix}}=\mathbf {e} _{i}\otimes \mathbf {I} _{m}}
966:
1591:
766:
1266:
598:
339:
351:
299:
942:
2566:
496:
2376:
2096:
86:
2444:
There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice versa called, respectively, the
2317:
437:
2037:) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the
910:{\displaystyle \operatorname {vec} (\operatorname {ad} _{A}(X))=(I_{n}\otimes A-A^{\mathrm {T} }\otimes I_{n}){\text{vec}}(X)}
2792:
2656:
1506:
2012:{\displaystyle \operatorname {vec} (\mathbf {X} )=\sum _{i=1}^{n}\mathbf {e} _{i}\otimes \mathbf {X} \mathbf {e} _{i}}
714:
2821:
51:
20:
1887:{\displaystyle \operatorname {vec} (\mathbf {X} )=\sum _{i=1}^{n}\mathbf {B} _{i}\mathbf {X} \mathbf {e} _{i}}
2497:
1453:{\displaystyle \operatorname {tr} (A^{\dagger }B)=\operatorname {vec} (A)^{\dagger }\operatorname {vec} (B),}
2489:
1364:
1256:
424:{\displaystyle \mathbf {R} ^{m\times n}:=\mathbf {R} ^{m}\otimes \mathbf {R} ^{n}\cong \mathbf {R} ^{mn}}
316:
2839:"Matrix differential calculus with applications in the multivariate linear model and its diagnostics"
1333:{\displaystyle \operatorname {vec} (A\circ B)=\operatorname {vec} (A)\circ \operatorname {vec} (B).}
2876:
2837:
Liu, Shuangzhe; Leiva, Victor; Zhuang, Dan; Ma, Tiefeng; Figueroa-Zúñiga, Jorge I. (March 2022).
2038:
2696:
Macedo, H. D.; Oliveira, J. N. (2013). "Typing Linear
Algebra: A Biproduct-oriented Approach".
2520:
2508:
1348:
2780:
2666:
592:
271:
47:
2636:{\displaystyle \operatorname {vec} (ABC)=(A\otimes C^{\mathrm {T} })\operatorname {vec} (B)}
668:{\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {T} }\otimes A)\operatorname {vec} (B)}
1242:
920:
769:
43:
35:
8:
1461:
2460:
Programming languages that implement matrices may have easy means for vectorization. In
1360:
2723:
2705:
2449:
2445:
576:
2881:
2817:
2788:
1926:
588:
2850:
2727:
2715:
2027:
1230:
2762:"The R package 'sn': The Skew-Normal and Related Distributions such as the Skew-t"
2671:
2540:
952:
2785:
Hands-on Matrix
Algebra Using R: Active and Motivated Learning with Applications
2719:
1472:
The matrix vectorization operation can be written in terms of a linear sum. Let
561:{\displaystyle \operatorname {vec} (A)={\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}}
2855:
2838:
2661:
796:
31:
2434:{\displaystyle \operatorname {vech} (A)={\begin{bmatrix}a\\b\\d\end{bmatrix}}}
2060:
is sometimes more useful than the vectorization. The half-vectorization, vech(
2870:
2814:
Matrix differential calculus with applications in statistics and econometrics
2676:
2053:
1368:
2798:
1777:, stacked column-wise, and all these matrices are all-zero except for the
773:
346:
27:
2761:
2742:
2089:
column vector obtained by vectorizing only the lower triangular part of
2477:
2465:
2531:
implemented in both packages 'ks' and 'sn' allows half-vectorization.
342:
2710:
2366:{\displaystyle A={\begin{bmatrix}a&b\\b&d\end{bmatrix}}}
486:{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}
431:
between these (i.e., of matrices and vectors) as vector spaces.
2461:
1233:
in the monoidal closed structure of any category of matrices.
2500:
79:
column vector obtained by stacking the columns of the matrix
1229:
More generally, it has been shown that vectorization is a
2305:{\displaystyle \operatorname {vech} (A)=^{\mathrm {T} }.}
1925:
Alternatively, the linear sum can be expressed using the
582:
1922:
puts it into the desired position in the final vector.
1236:
595:
as a linear transformation on matrices. In particular,
587:
The vectorization is frequently used together with the
575:
and the vectorization of its transpose is given by the
261:{\displaystyle \operatorname {vec} (A)=^{\mathrm {T} }}
2480:
also allows vectorization and half-vectorization with
2403:
2332:
1640:
1509:
523:
452:
2569:
2379:
2320:
2099:
1935:
1813:
1619:
1381:
1269:
964:
923:
805:
717:
601:
499:
440:
354:
319:
274:
89:
1342:
345:. Vectorization expresses, through coordinates, the
2635:
2527:of package 'ks' allows vectorization and function
2433:
2365:
2304:
2011:
1886:
1745:
1586:{\textstyle \mathbf {e} _{i}=\left^{\mathrm {T} }}
1585:
1452:
1332:
1219:
936:
909:
760:
667:
560:
485:
423:
333:
293:
260:
2787:. Singapore: World Scientific. pp. 233–248.
2868:
2836:
1467:
761:{\displaystyle \operatorname {ad} _{A}(X)=AX-XA}
2811:
2695:
16:Conversion of a matrix or a tensor to a vector
2563:The identity for row-major vectorization is
571:The connection between the vectorization of
2511:the desired effect can be achieved via the
2559:
2557:
1486:matrix that we want to vectorize, and let
2854:
2781:"Simultaneous Reduction and Vec Stacking"
2709:
958:There are two other useful formulations:
2759:
2554:
2455:
54:. Specifically, the vectorization of a
2869:
2812:Magnus, Jan; Neudecker, Heinz (2019).
2778:
2740:
2021:
583:Compatibility with Kronecker products
2657:Duplication and elimination matrices
1913:-th column, while multiplication by
1237:Compatibility with Hadamard products
1499:-th canonical basis vector for the
1460:where the superscript denotes the
13:
2609:
2293:
1577:
1176:
1061:
1049:
871:
635:
325:
252:
14:
2893:
1613:block matrix defined as follows:
1343:Compatibility with inner products
334:{\displaystyle {}^{\mathrm {T} }}
50:which converts the matrix into a
2843:Journal of Multivariate Analysis
2314:For example, for the 2×2 matrix
1999:
1993:
1979:
1946:
1874:
1868:
1857:
1824:
1733:
1718:
1701:
1685:
1670:
1660:
1644:
1622:
1512:
434:For example, for the 2×2 matrix
408:
393:
378:
357:
2698:Science of Computer Programming
2534:
1803:Then the vectorized version of
2830:
2805:
2772:
2753:
2734:
2689:
2630:
2624:
2615:
2594:
2588:
2576:
2392:
2386:
2288:
2118:
2112:
2106:
1950:
1942:
1828:
1820:
1444:
1438:
1423:
1416:
1404:
1388:
1324:
1318:
1306:
1300:
1288:
1276:
1210:
1204:
1195:
1167:
1161:
1155:
1146:
1127:
1117:
1108:
1095:
1089:
1080:
1040:
1034:
1028:
1019:
997:
987:
975:
904:
898:
890:
843:
837:
834:
828:
812:
737:
731:
662:
656:
647:
626:
620:
608:
512:
506:
301:represents the element in the
247:
108:
102:
96:
1:
2779:Vinod, Hrishikesh D. (2011).
2682:
1807:can be expressed as follows:
1468:Vectorization as a linear sum
2373:, the half-vectorization is
1503:-dimensional space, that is
7:
2720:10.1016/j.scico.2012.07.012
2650:
1263:with its Hadamard product:
10:
2898:
2856:10.1016/j.jmva.2021.104849
2760:Azzalini, Adelchi (2017).
18:
2539:Vectorization is used in
2056:. For such matrices, the
2052:entries on and below the
2816:. New York: John Wiley.
2747:R package version 1.11.0
2547:
2766:R package version 1.5.1
1767:block matrices of size
1259:(entrywise) product to
493:, the vectorization is
294:{\displaystyle a_{i,j}}
83:on top of one another:
2743:"ks: Kernel Smoothing"
2637:
2435:
2367:
2306:
2041:portion, that is, the
2013:
1976:
1888:
1854:
1747:
1587:
1454:
1349:unitary transformation
1334:
1221:
938:
911:
762:
669:
562:
487:
425:
335:
313:, and the superscript
295:
262:
2667:Packed storage matrix
2638:
2503:arrays implement the
2496:function as well. In
2472:can be vectorized by
2436:
2368:
2307:
2014:
1956:
1889:
1834:
1748:
1588:
1455:
1335:
1222:
939:
937:{\displaystyle I_{n}}
912:
763:
670:
593:matrix multiplication
563:
488:
426:
336:
296:
263:
48:linear transformation
2741:Duong, Tarn (2018).
2567:
2456:Programming language
2377:
2318:
2097:
1933:
1811:
1781:-th one, which is a
1617:
1507:
1379:
1267:
1243:algebra homomorphism
1241:Vectorization is an
962:
921:
803:
770:adjoint endomorphism
715:
599:
497:
438:
352:
317:
272:
87:
19:For other uses, see
1462:conjugate transpose
1347:Vectorization is a
2672:Column-major order
2633:
2450:elimination matrix
2446:duplication matrix
2431:
2425:
2363:
2357:
2302:
2064:), of a symmetric
2058:half-vectorization
2022:Half-vectorization
2009:
1896:Multiplication of
1884:
1743:
1707:
1583:
1450:
1359:matrices with the
1351:from the space of
1330:
1255:matrices with the
1245:from the space of
1217:
1215:
934:
907:
758:
711:. For example, if
665:
577:commutation matrix
558:
552:
483:
477:
421:
331:
291:
258:
2794:978-981-4313-69-8
2704:(11): 2160–2191.
2507:method, while in
2033:, the vector vec(
1927:Kronecker product
896:
589:Kronecker product
2889:
2861:
2860:
2858:
2834:
2828:
2827:
2809:
2803:
2802:
2776:
2770:
2769:
2757:
2751:
2750:
2738:
2732:
2731:
2713:
2693:
2644:
2642:
2640:
2639:
2634:
2614:
2613:
2612:
2561:
2530:
2526:
2518:
2514:
2506:
2495:
2487:
2483:
2475:
2471:
2440:
2438:
2437:
2432:
2430:
2429:
2372:
2370:
2369:
2364:
2362:
2361:
2311:
2309:
2308:
2303:
2298:
2297:
2296:
2286:
2285:
2267:
2266:
2242:
2241:
2205:
2204:
2180:
2179:
2161:
2160:
2136:
2135:
2088:
2073:
2051:
2039:lower triangular
2028:symmetric matrix
2018:
2016:
2015:
2010:
2008:
2007:
2002:
1996:
1988:
1987:
1982:
1975:
1970:
1949:
1893:
1891:
1890:
1885:
1883:
1882:
1877:
1871:
1866:
1865:
1860:
1853:
1848:
1827:
1791:identity matrix
1790:
1776:
1752:
1750:
1749:
1744:
1742:
1741:
1736:
1727:
1726:
1721:
1712:
1711:
1704:
1688:
1679:
1678:
1673:
1663:
1647:
1631:
1630:
1625:
1612:
1592:
1590:
1589:
1584:
1582:
1581:
1580:
1574:
1570:
1521:
1520:
1515:
1485:
1459:
1457:
1456:
1451:
1431:
1430:
1400:
1399:
1339:
1337:
1336:
1331:
1254:
1226:
1224:
1223:
1218:
1216:
1194:
1193:
1181:
1180:
1179:
1139:
1138:
1079:
1078:
1066:
1065:
1064:
1054:
1053:
1052:
1009:
1008:
943:
941:
940:
935:
933:
932:
916:
914:
913:
908:
897:
894:
889:
888:
876:
875:
874:
855:
854:
824:
823:
786:
767:
765:
764:
759:
727:
726:
674:
672:
671:
666:
640:
639:
638:
567:
565:
564:
559:
557:
556:
492:
490:
489:
484:
482:
481:
430:
428:
427:
422:
420:
419:
411:
402:
401:
396:
387:
386:
381:
372:
371:
360:
340:
338:
337:
332:
330:
329:
328:
322:
300:
298:
297:
292:
290:
289:
267:
265:
264:
259:
257:
256:
255:
245:
244:
220:
219:
195:
194:
170:
169:
151:
150:
126:
125:
78:
63:
30:, especially in
2897:
2896:
2892:
2891:
2890:
2888:
2887:
2886:
2867:
2866:
2865:
2864:
2835:
2831:
2824:
2810:
2806:
2795:
2777:
2773:
2758:
2754:
2739:
2735:
2694:
2690:
2685:
2653:
2648:
2647:
2608:
2607:
2603:
2568:
2565:
2564:
2562:
2555:
2550:
2541:matrix calculus
2537:
2528:
2524:
2516:
2512:
2504:
2493:
2485:
2481:
2473:
2469:
2458:
2424:
2423:
2417:
2416:
2410:
2409:
2399:
2398:
2378:
2375:
2374:
2356:
2355:
2350:
2344:
2343:
2338:
2328:
2327:
2319:
2316:
2315:
2292:
2291:
2287:
2275:
2271:
2250:
2246:
2219:
2215:
2194:
2190:
2169:
2165:
2150:
2146:
2125:
2121:
2098:
2095:
2094:
2079:
2065:
2042:
2024:
2003:
1998:
1997:
1992:
1983:
1978:
1977:
1971:
1960:
1945:
1934:
1931:
1930:
1921:
1908:
1878:
1873:
1872:
1867:
1861:
1856:
1855:
1849:
1838:
1823:
1812:
1809:
1808:
1799:
1782:
1768:
1762:
1737:
1732:
1731:
1722:
1717:
1716:
1706:
1705:
1700:
1697:
1696:
1690:
1689:
1684:
1681:
1680:
1674:
1669:
1668:
1665:
1664:
1659:
1656:
1655:
1649:
1648:
1643:
1636:
1635:
1626:
1621:
1620:
1618:
1615:
1614:
1603:
1601:
1576:
1575:
1530:
1526:
1525:
1516:
1511:
1510:
1508:
1505:
1504:
1494:
1477:
1470:
1426:
1422:
1395:
1391:
1380:
1377:
1376:
1365:Hilbert–Schmidt
1345:
1268:
1265:
1264:
1246:
1239:
1231:self-adjunction
1214:
1213:
1189:
1185:
1175:
1174:
1170:
1134:
1130:
1120:
1099:
1098:
1074:
1070:
1060:
1059:
1055:
1048:
1047:
1043:
1004:
1000:
990:
965:
963:
960:
959:
953:identity matrix
928:
924:
922:
919:
918:
893:
884:
880:
870:
869:
865:
850:
846:
819:
815:
804:
801:
800:
799:entries), then
776:
722:
718:
716:
713:
712:
634:
633:
629:
600:
597:
596:
585:
551:
550:
544:
543:
537:
536:
530:
529:
519:
518:
498:
495:
494:
476:
475:
470:
464:
463:
458:
448:
447:
439:
436:
435:
412:
407:
406:
397:
392:
391:
382:
377:
376:
361:
356:
355:
353:
350:
349:
324:
323:
321:
320:
318:
315:
314:
279:
275:
273:
270:
269:
251:
250:
246:
234:
230:
209:
205:
184:
180:
159:
155:
140:
136:
115:
111:
88:
85:
84:
73:
55:
24:
17:
12:
11:
5:
2895:
2885:
2884:
2879:
2877:Linear algebra
2863:
2862:
2829:
2822:
2804:
2793:
2771:
2752:
2733:
2687:
2686:
2684:
2681:
2680:
2679:
2674:
2669:
2664:
2662:Voigt notation
2659:
2652:
2649:
2646:
2645:
2632:
2629:
2626:
2623:
2620:
2617:
2611:
2606:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2552:
2551:
2549:
2546:
2536:
2533:
2519:functions. In
2488:respectively.
2457:
2454:
2428:
2422:
2419:
2418:
2415:
2412:
2411:
2408:
2405:
2404:
2402:
2397:
2394:
2391:
2388:
2385:
2382:
2360:
2354:
2351:
2349:
2346:
2345:
2342:
2339:
2337:
2334:
2333:
2331:
2326:
2323:
2301:
2295:
2290:
2284:
2281:
2278:
2274:
2270:
2265:
2262:
2259:
2256:
2253:
2249:
2245:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2218:
2214:
2211:
2208:
2203:
2200:
2197:
2193:
2189:
2186:
2183:
2178:
2175:
2172:
2168:
2164:
2159:
2156:
2153:
2149:
2145:
2142:
2139:
2134:
2131:
2128:
2124:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2023:
2020:
2006:
2001:
1995:
1991:
1986:
1981:
1974:
1969:
1966:
1963:
1959:
1955:
1952:
1948:
1944:
1941:
1938:
1917:
1904:
1881:
1876:
1870:
1864:
1859:
1852:
1847:
1844:
1841:
1837:
1833:
1830:
1826:
1822:
1819:
1816:
1795:
1758:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1703:
1699:
1698:
1695:
1692:
1691:
1687:
1683:
1682:
1677:
1672:
1667:
1666:
1662:
1658:
1657:
1654:
1651:
1650:
1646:
1642:
1641:
1639:
1634:
1629:
1624:
1597:
1579:
1573:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1529:
1524:
1519:
1514:
1490:
1469:
1466:
1449:
1446:
1443:
1440:
1437:
1434:
1429:
1425:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1398:
1394:
1390:
1387:
1384:
1344:
1341:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1238:
1235:
1212:
1209:
1206:
1203:
1200:
1197:
1192:
1188:
1184:
1178:
1173:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1137:
1133:
1129:
1126:
1123:
1121:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1077:
1073:
1069:
1063:
1058:
1051:
1046:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1007:
1003:
999:
996:
993:
991:
989:
986:
983:
980:
977:
974:
971:
968:
967:
931:
927:
906:
903:
900:
892:
887:
883:
879:
873:
868:
864:
861:
858:
853:
849:
845:
842:
839:
836:
833:
830:
827:
822:
818:
814:
811:
808:
795:matrices with
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
725:
721:
687:of dimensions
664:
661:
658:
655:
652:
649:
646:
643:
637:
632:
628:
625:
622:
619:
616:
613:
610:
607:
604:
584:
581:
555:
549:
546:
545:
542:
539:
538:
535:
532:
531:
528:
525:
524:
522:
517:
514:
511:
508:
505:
502:
480:
474:
471:
469:
466:
465:
462:
459:
457:
454:
453:
451:
446:
443:
418:
415:
410:
405:
400:
395:
390:
385:
380:
375:
370:
367:
364:
359:
327:
309:-th column of
288:
285:
282:
278:
254:
249:
243:
240:
237:
233:
229:
226:
223:
218:
215:
212:
208:
204:
201:
198:
193:
190:
187:
183:
179:
176:
173:
168:
165:
162:
158:
154:
149:
146:
143:
139:
135:
132:
129:
124:
121:
118:
114:
110:
107:
104:
101:
98:
95:
92:
68:, denoted vec(
32:linear algebra
15:
9:
6:
4:
3:
2:
2894:
2883:
2880:
2878:
2875:
2874:
2872:
2857:
2852:
2848:
2844:
2840:
2833:
2825:
2823:9781119541202
2819:
2815:
2808:
2800:
2796:
2790:
2786:
2782:
2775:
2767:
2763:
2756:
2748:
2744:
2737:
2729:
2725:
2721:
2717:
2712:
2707:
2703:
2699:
2692:
2688:
2678:
2677:Matricization
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2654:
2627:
2621:
2618:
2604:
2600:
2597:
2591:
2585:
2582:
2579:
2573:
2570:
2560:
2558:
2553:
2545:
2542:
2532:
2522:
2510:
2502:
2499:
2491:
2479:
2467:
2463:
2453:
2451:
2447:
2442:
2426:
2420:
2413:
2406:
2400:
2395:
2389:
2383:
2380:
2358:
2352:
2347:
2340:
2335:
2329:
2324:
2321:
2312:
2299:
2282:
2279:
2276:
2272:
2268:
2263:
2260:
2257:
2254:
2251:
2247:
2243:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2216:
2212:
2209:
2206:
2201:
2198:
2195:
2191:
2187:
2184:
2181:
2176:
2173:
2170:
2166:
2162:
2157:
2154:
2151:
2147:
2143:
2140:
2137:
2132:
2129:
2126:
2122:
2115:
2109:
2103:
2100:
2092:
2086:
2082:
2077:
2072:
2068:
2063:
2059:
2055:
2054:main diagonal
2049:
2045:
2040:
2036:
2032:
2029:
2019:
2004:
1989:
1984:
1972:
1967:
1964:
1961:
1957:
1953:
1939:
1936:
1928:
1923:
1920:
1916:
1912:
1909:extracts the
1907:
1903:
1899:
1894:
1879:
1862:
1850:
1845:
1842:
1839:
1835:
1831:
1817:
1814:
1806:
1801:
1798:
1794:
1789:
1785:
1780:
1775:
1771:
1766:
1761:
1757:
1753:
1738:
1728:
1723:
1713:
1708:
1693:
1675:
1652:
1637:
1632:
1627:
1611:
1607:
1600:
1596:
1571:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1527:
1522:
1517:
1502:
1498:
1493:
1489:
1484:
1480:
1475:
1465:
1463:
1447:
1441:
1435:
1432:
1427:
1419:
1413:
1410:
1407:
1401:
1396:
1392:
1385:
1382:
1374:
1370:
1369:inner product
1366:
1362:
1358:
1354:
1350:
1340:
1327:
1321:
1315:
1312:
1309:
1303:
1297:
1294:
1291:
1285:
1282:
1279:
1273:
1270:
1262:
1258:
1253:
1249:
1244:
1234:
1232:
1227:
1207:
1201:
1198:
1190:
1186:
1182:
1171:
1164:
1158:
1152:
1149:
1143:
1140:
1135:
1131:
1124:
1122:
1114:
1111:
1105:
1102:
1092:
1086:
1083:
1075:
1071:
1067:
1056:
1044:
1037:
1031:
1025:
1022:
1016:
1013:
1010:
1005:
1001:
994:
992:
984:
981:
978:
972:
969:
956:
954:
951:
947:
929:
925:
901:
885:
881:
877:
866:
862:
859:
856:
851:
847:
840:
831:
825:
820:
816:
809:
806:
798:
794:
790:
784:
780:
775:
771:
755:
752:
749:
746:
743:
740:
734:
728:
723:
719:
710:
706:
702:
698:
694:
690:
686:
682:
678:
675:for matrices
659:
653:
650:
644:
641:
630:
623:
617:
614:
611:
605:
602:
594:
590:
580:
578:
574:
569:
553:
547:
540:
533:
526:
520:
515:
509:
503:
500:
478:
472:
467:
460:
455:
449:
444:
441:
432:
416:
413:
403:
398:
388:
383:
373:
368:
365:
362:
348:
344:
312:
308:
304:
286:
283:
280:
276:
241:
238:
235:
231:
227:
224:
221:
216:
213:
210:
206:
202:
199:
196:
191:
188:
185:
181:
177:
174:
171:
166:
163:
160:
156:
152:
147:
144:
141:
137:
133:
130:
127:
122:
119:
116:
112:
105:
99:
93:
90:
82:
76:
71:
67:
62:
58:
53:
49:
45:
41:
40:vectorization
37:
36:matrix theory
33:
29:
22:
21:Vectorization
2846:
2842:
2832:
2813:
2807:
2799:Google Books
2797:– via
2784:
2774:
2765:
2755:
2746:
2736:
2701:
2697:
2691:
2538:
2535:Applications
2459:
2443:
2313:
2090:
2084:
2080:
2075:
2070:
2066:
2061:
2057:
2047:
2043:
2034:
2030:
2025:
1924:
1918:
1914:
1910:
1905:
1901:
1897:
1895:
1804:
1802:
1796:
1792:
1787:
1783:
1778:
1773:
1769:
1764:
1763:consists of
1759:
1755:
1754:
1609:
1605:
1598:
1594:
1500:
1496:
1491:
1487:
1482:
1478:
1473:
1471:
1372:
1356:
1352:
1346:
1260:
1251:
1247:
1240:
1228:
957:
949:
945:
792:
788:
782:
778:
708:
704:
700:
696:
692:
688:
684:
680:
676:
586:
572:
570:
433:
341:denotes the
310:
306:
305:-th row and
302:
80:
74:
69:
65:
60:
56:
39:
25:
2523:, function
2517:as.vector()
774:Lie algebra
591:to express
347:isomorphism
28:mathematics
2871:Categories
2849:: 104849.
2683:References
2478:GNU Octave
2466:GNU Octave
2087:+ 1)/2 × 1
72:), is the
2711:1312.4818
2622:
2601:⊗
2574:
2468:a matrix
2384:
2261:−
2236:−
2224:−
2210:…
2185:…
2141:…
2104:
1990:⊗
1958:∑
1940:
1836:∑
1818:
1729:⊗
1694:⋮
1653:⋮
1562:…
1538:…
1436:
1428:†
1414:
1397:†
1386:
1361:Frobenius
1316:
1310:∘
1298:
1283:∘
1274:
1202:
1183:⊗
1153:
1141:⊗
1106:
1087:
1068:⊗
1026:
1011:⊗
973:
878:⊗
863:−
857:⊗
826:
810:
750:−
729:
654:
642:⊗
606:
504:
404:≅
389:⊗
366:×
343:transpose
225:…
200:…
175:…
131:…
94:
2882:Matrices
2651:See also
2492:has the
2448:and the
1257:Hadamard
917:, where
2728:9846072
2505:flatten
2486:vech(A)
2078:is the
2074:matrix
1495:be the
944:is the
797:complex
787:of all
772:of the
64:matrix
2820:
2791:
2726:
2529:vech()
2498:Python
2494:vec(A)
2482:vec(A)
2462:Matlab
2050:+ 1)/2
2026:For a
1593:. Let
1476:be an
703:, and
683:, and
268:Here,
52:vector
44:matrix
38:, the
2724:S2CID
2706:arXiv
2548:Notes
2525:vec()
2501:NumPy
2490:Julia
1602:be a
768:(the
46:is a
42:of a
2818:ISBN
2789:ISBN
2484:and
2474:A(:)
2381:vech
2101:vech
1608:) ×
1363:(or
34:and
2851:doi
2847:188
2716:doi
2619:vec
2571:vec
2515:or
2513:c()
1937:vec
1900:by
1815:vec
1433:vec
1411:vec
1371:to
1313:vec
1295:vec
1271:vec
1199:vec
1150:vec
1103:vec
1084:vec
1023:vec
970:vec
895:vec
807:vec
777:gl(
651:vec
603:vec
501:vec
91:vec
77:× 1
26:In
2873::
2845:.
2841:.
2783:.
2764:.
2745:.
2722:.
2714:.
2702:78
2700:.
2556:^
2476:.
2452:.
2441:.
2093::
2069:×
1929::
1800:.
1786:×
1772:×
1606:mn
1481:×
1464:.
1383:tr
1375::
1367:)
1250:×
955:.
817:ad
781:,
720:ad
695:,
679:,
579:.
568:.
374::=
75:mn
59:×
2859:.
2853::
2826:.
2801:.
2768:.
2749:.
2730:.
2718::
2708::
2643:.
2631:)
2628:B
2625:(
2616:)
2610:T
2605:C
2598:A
2595:(
2592:=
2589:)
2586:C
2583:B
2580:A
2577:(
2521:R
2509:R
2470:A
2464:/
2427:]
2421:d
2414:b
2407:a
2401:[
2396:=
2393:)
2390:A
2387:(
2359:]
2353:d
2348:b
2341:b
2336:a
2330:[
2325:=
2322:A
2300:.
2294:T
2289:]
2283:n
2280:,
2277:n
2273:A
2269:,
2264:1
2258:n
2255:,
2252:n
2248:A
2244:,
2239:1
2233:n
2230:,
2227:1
2221:n
2217:A
2213:,
2207:,
2202:2
2199:,
2196:n
2192:A
2188:,
2182:,
2177:2
2174:,
2171:2
2167:A
2163:,
2158:1
2155:,
2152:n
2148:A
2144:,
2138:,
2133:1
2130:,
2127:1
2123:A
2119:[
2116:=
2113:)
2110:A
2107:(
2091:A
2085:n
2083:(
2081:n
2076:A
2071:n
2067:n
2062:A
2048:n
2046:(
2044:n
2035:A
2031:A
2005:i
2000:e
1994:X
1985:i
1980:e
1973:n
1968:1
1965:=
1962:i
1954:=
1951:)
1947:X
1943:(
1919:i
1915:B
1911:i
1906:i
1902:e
1898:X
1880:i
1875:e
1869:X
1863:i
1858:B
1851:n
1846:1
1843:=
1840:i
1832:=
1829:)
1825:X
1821:(
1805:X
1797:m
1793:I
1788:m
1784:m
1779:i
1774:m
1770:m
1765:n
1760:i
1756:B
1739:m
1734:I
1724:i
1719:e
1714:=
1709:]
1702:0
1686:0
1676:m
1671:I
1661:0
1645:0
1638:[
1633:=
1628:i
1623:B
1610:m
1604:(
1599:i
1595:B
1578:T
1572:]
1568:0
1565:,
1559:,
1556:0
1553:,
1550:1
1547:,
1544:0
1541:,
1535:,
1532:0
1528:[
1523:=
1518:i
1513:e
1501:n
1497:i
1492:i
1488:e
1483:n
1479:m
1474:X
1448:,
1445:)
1442:B
1439:(
1424:)
1420:A
1417:(
1408:=
1405:)
1402:B
1393:A
1389:(
1373:C
1357:n
1355:×
1353:n
1328:.
1325:)
1322:B
1319:(
1307:)
1304:A
1301:(
1292:=
1289:)
1286:B
1280:A
1277:(
1261:C
1252:n
1248:n
1211:)
1208:A
1205:(
1196:)
1191:k
1187:I
1177:T
1172:B
1168:(
1165:=
1162:)
1159:B
1156:(
1147:)
1144:A
1136:m
1132:I
1128:(
1125:=
1118:)
1115:B
1112:A
1109:(
1096:)
1093:A
1090:(
1081:)
1076:k
1072:I
1062:T
1057:B
1050:T
1045:C
1041:(
1038:=
1035:)
1032:C
1029:(
1020:)
1017:B
1014:A
1006:n
1002:I
998:(
995:=
988:)
985:C
982:B
979:A
976:(
950:n
948:×
946:n
930:n
926:I
905:)
902:X
899:(
891:)
886:n
882:I
872:T
867:A
860:A
852:n
848:I
844:(
841:=
838:)
835:)
832:X
829:(
821:A
813:(
793:n
791:×
789:n
785:)
783:C
779:n
756:A
753:X
747:X
744:A
741:=
738:)
735:X
732:(
724:A
709:n
707:×
705:m
701:m
699:×
697:l
693:l
691:×
689:k
685:C
681:B
677:A
663:)
660:B
657:(
648:)
645:A
636:T
631:C
627:(
624:=
621:)
618:C
615:B
612:A
609:(
573:A
554:]
548:d
541:b
534:c
527:a
521:[
516:=
513:)
510:A
507:(
479:]
473:d
468:c
461:b
456:a
450:[
445:=
442:A
417:n
414:m
409:R
399:n
394:R
384:m
379:R
369:n
363:m
358:R
326:T
311:A
307:j
303:i
287:j
284:,
281:i
277:a
253:T
248:]
242:n
239:,
236:m
232:a
228:,
222:,
217:n
214:,
211:1
207:a
203:,
197:,
192:2
189:,
186:m
182:a
178:,
172:,
167:2
164:,
161:1
157:a
153:,
148:1
145:,
142:m
138:a
134:,
128:,
123:1
120:,
117:1
113:a
109:[
106:=
103:)
100:A
97:(
81:A
70:A
66:A
61:n
57:m
23:.
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