640:
400:
1568:
870:). The converse is also true; that is, if two diagonalizable matrices commute, they are simultaneously diagonalizable. But if you take any two matrices that commute (and do not assume they are two diagonalizable matrices) they are simultaneously diagonalizable already if one of the matrices has no multiple eigenvalues.
2190:
635:{\displaystyle {\begin{bmatrix}1&2\\0&3\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}1&3\\0&3\end{bmatrix}}\neq {\begin{bmatrix}1&5\\0&3\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&2\\0&3\end{bmatrix}}.}
1368:
1950:
1682:
commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to
Hermitian matrices without multiple eigenvalues, then commutativity is transitive, as a consequence of the characterization in terms of
2472:. In each block, the first matrix is the product of an identity matrix, and the second one is a diagonalizable matrix. So, diagonalizing the blocks of the second matrix does change the first matrix, and allows a simultaneous diagonalization.
1190:
coincide. In particular, two
Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. This follows by considering the eigenvalue decompositions of both matrices. Let
1563:{\displaystyle AB=U\Lambda _{1}U^{\dagger }U\Lambda _{2}U^{\dagger }=U\Lambda _{1}\Lambda _{2}U^{\dagger }=U\Lambda _{2}\Lambda _{1}U^{\dagger }=U\Lambda _{2}U^{\dagger }U\Lambda _{1}U^{\dagger }=BA.}
1089:
1361:
1315:
1173:
389:
282:
203:
2466:
1941:
1857:
2185:{\displaystyle 1+\sum _{n=1}^{\infty }{\frac {P(n)}{(q^{n}-1)(q^{n}-q)\cdots (q^{n}-q^{n-1})}}z^{n}=\prod _{i=1}^{\infty }\prod _{j=0}^{\infty }{\frac {1}{1-q^{1-j}z^{i}}}.}
345:
2378:
1912:
1757:
1805:
864:
828:
2411:
2237:
in his memoir on the theory of matrices, which also provided the first axiomatization of matrices. The first significant results on commuting matrices were proved by
1124:
691:
153:
768:
100:
1886:
1777:
1726:
1676:
1656:
1636:
1616:
1596:
1269:
1249:
1229:
1209:
1011:
991:
971:
951:
931:
911:
891:
792:
736:
716:
302:
64:
44:
1021:
2214:
is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices.
1054:
2317:
2292:
2267:
1020:(that is, it has the maximal degree), which happens in particular whenever the characteristic polynomial has only
1320:
1274:
227:
1129:
771:
350:
241:
162:
2332:
1690:
223:
2416:
1917:
1833:
2238:
1176:
1017:
307:
2482:
2338:
235:
2642:
1891:
1739:
1044:
231:
1790:
833:
797:
2383:
1808:
1094:
651:
108:
1694:
741:
73:
24:
1862:
8:
1812:
1575:
2586:
2217:
1762:
1711:
1661:
1641:
1621:
1601:
1581:
1254:
1234:
1214:
1194:
1016:
If one of the matrices has the property that its minimal polynomial coincides with its
996:
976:
956:
936:
916:
896:
876:
777:
721:
701:
287:
49:
29:
2546:
2313:
2288:
2263:
156:
2619:
2538:
2526:
2221:
2211:
1686:
1183:
392:
209:
if they commute pairwise, meaning that every pair of matrices in the set commutes.
2542:
1679:
867:
2468:
After a permutation of rows and columns, the two matrices become simultaneously
2469:
1032:
20:
2623:
2636:
2550:
2234:
2207:
commute with upper triangular matrices that have the same value along bands.
2564:
2506:
Frobenius, G. (1877). "Ueber lineare
Substitutionen und bilineare Formen".
2204:
1736:
matrix if and only if it is a scalar matrix, that is, a matrix of the form
1697:
is simultaneously upper triangularizable may be viewed as a generalization.
648:
However, if the square of the commutator of two matrices is zero, that is,
1944:
1187:
1028:
219:
103:
1024:, then the other matrix can be written as a polynomial in the first.
2610:
1824:
1051:
of roots of their characteristic polynomials) can be matched up as
1048:
2380:
is diagonal. In this case, commutativity implies that if an entry
1091:
in such a way that the multiset of eigenvalues of any polynomial
395:
is not necessarily true, as the following counterexample shows:
1027:
As a direct consequence of simultaneous triangulizability, the
598:
562:
523:
484:
445:
409:
2419:
2386:
2341:
2224:
since the sum of two circulant matrices is circulant.
1953:
1920:
1894:
1865:
1836:
1793:
1765:
1742:
1714:
1664:
1644:
1624:
1604:
1584:
1371:
1323:
1277:
1257:
1237:
1217:
1197:
1132:
1097:
1084:{\displaystyle \alpha _{i}\leftrightarrow \beta _{i}}
1057:
999:
979:
959:
939:
919:
899:
879:
836:
800:
780:
744:
724:
704:
654:
403:
353:
310:
290:
244:
165:
111:
76:
52:
32:
1678:
do not commute with each other. As an example, the
1271:
have common eigenspaces when they can be written as
2233:The notion of commuting matrices was introduced by
212:
2460:
2405:
2372:
2184:
1935:
1906:
1880:
1851:
1799:
1771:
1751:
1720:
1670:
1650:
1630:
1610:
1590:
1562:
1355:
1309:
1263:
1243:
1223:
1203:
1167:
1126:in the two matrices is the multiset of the values
1118:
1083:
1005:
985:
965:
945:
925:
905:
885:
858:
822:
786:
762:
730:
710:
685:
634:
383:
339:
296:
276:
197:
147:
94:
58:
38:
2527:"Pairs of commuting matrices over a finite field"
2634:
1888:denote the number of ordered pairs of commuting
2508:Journal für die reine und angewandte Mathematik
2483:"Proofs Homework Set 10 MATH 217 — WINTER 2011"
2201:The identity matrix commutes with all matrices.
222:. As a consequence, commuting matrices over an
2612:Proceedings of the London Mathematical Society
1574:The property of two matrices commuting is not
2308:Horn, Roger A.; Johnson, Charles R. (2013).
2283:Horn, Roger A.; Johnson, Charles R. (2012).
2258:Horn, Roger A.; Johnson, Charles R. (2012).
913:commute, they have a common eigenvector. If
774:(that is, there exists an invertible matrix
378:
360:
2307:
2287:. Cambridge University Press. p. 127.
2282:
2257:
1356:{\displaystyle B=U\Lambda _{2}U^{\dagger }}
1310:{\displaystyle A=U\Lambda _{1}U^{\dagger }}
2301:
2262:. Cambridge University Press. p. 70.
284:commute, there exists a similarity matrix
2505:
1923:
1839:
1168:{\displaystyle P(\alpha _{i},\beta _{i})}
218:Commuting matrices preserve each other's
2525:Feit, Walter; Fine, N. J. (1960-03-01).
2524:
2335:, one may suppose that the first matrix
2635:
2609:
2565:"Do Diagonal Matrices Always Commute?"
2413:of the second matrix is nonzero, then
1823:matrices under multiplication is the
384:{\displaystyle i\in \{1,\ldots ,k\}}
277:{\displaystyle A_{1},\ldots ,A_{k}}
198:{\displaystyle A_{1},\ldots ,A_{k}}
13:
2133:
2112:
1976:
1947:and N. J. Fine showed the equation
1529:
1506:
1480:
1470:
1444:
1434:
1408:
1385:
1334:
1288:
14:
2654:
1807:is a scalar. In other words, the
2587:"Linear Algebra WebNotes part 2"
2567:. Stack Exchange. March 15, 2016
2461:{\displaystyle a_{i,i}=a_{j,j}.}
1936:{\displaystyle \mathbb {F} _{q}}
1852:{\displaystyle \mathbb {F} _{q}}
213:Characterizations and properties
2603:
2310:Matrix Analysis, second edition
228:simultaneously triangularizable
16:Mathematical concept in algebra
2579:
2557:
2518:
2499:
2475:
2367:
2348:
2326:
2312:. Cambridge University Press.
2276:
2251:
2077:
2045:
2039:
2020:
2017:
1998:
1993:
1987:
1875:
1869:
1162:
1136:
1113:
1101:
1068:
933:has distinct eigenvalues, and
668:
655:
124:
112:
1:
2543:10.1215/s0012-7094-60-02709-5
2244:
772:simultaneously diagonalizable
1231:be two Hermitian matrices.
698:Two diagonalizable matrices
693:, then the converse is true.
347:is upper triangular for all
340:{\displaystyle P^{-1}A_{i}P}
7:
2373:{\displaystyle A=(a_{i,j})}
2195:
102:, or equivalently if their
10:
2659:
2333:Without loss of generality
2228:
1728:commutes with every other
1186:matrices commute if their
224:algebraically closed field
2531:Duke Mathematical Journal
1907:{\displaystyle n\times n}
1752:{\displaystyle \lambda I}
1175:. This theorem is due to
1018:characteristic polynomial
234:over which they are both
1800:{\displaystyle \lambda }
1363:. It then follows that
1045:algebraic multiplicities
859:{\displaystyle P^{-1}BP}
823:{\displaystyle P^{-1}AP}
2624:10.1112/plms/s3-1.1.222
2406:{\displaystyle b_{i,j}}
1689:, which shows that any
2462:
2407:
2374:
2210:If the product of two
2186:
2137:
2116:
1980:
1937:
1908:
1882:
1853:
1801:
1773:
1753:
1722:
1672:
1652:
1632:
1612:
1598:may commute with both
1592:
1564:
1357:
1311:
1265:
1245:
1225:
1205:
1169:
1120:
1119:{\displaystyle P(A,B)}
1085:
1007:
987:
967:
947:
927:
907:
887:
860:
824:
788:
764:
732:
712:
687:
686:{\displaystyle ^{2}=0}
636:
385:
341:
298:
278:
199:
149:
148:{\displaystyle =AB-BA}
96:
60:
40:
2463:
2408:
2375:
2220:commute. They form a
2187:
2117:
2096:
1960:
1938:
1909:
1883:
1854:
1802:
1774:
1754:
1723:
1673:
1653:
1633:
1613:
1593:
1565:
1358:
1312:
1266:
1246:
1226:
1206:
1170:
1121:
1086:
1008:
988:
968:
948:
928:
908:
888:
861:
825:
789:
765:
763:{\displaystyle AB=BA}
733:
713:
688:
637:
386:
342:
299:
279:
238:. In other words, if
230:; that is, there are
200:
150:
97:
95:{\displaystyle AB=BA}
61:
41:
2417:
2384:
2339:
1951:
1918:
1892:
1881:{\displaystyle P(n)}
1863:
1834:
1791:
1787:identity matrix and
1763:
1740:
1712:
1695:solvable Lie algebra
1662:
1642:
1622:
1602:
1582:
1369:
1321:
1275:
1255:
1235:
1215:
1195:
1130:
1095:
1055:
997:
993:'s eigenvectors are
977:
957:
937:
917:
897:
877:
834:
798:
778:
742:
722:
702:
652:
401:
351:
308:
288:
242:
163:
109:
74:
50:
30:
2591:math.vanderbilt.edu
1830:Fix a finite field
1827:of scalar matrices.
2458:
2403:
2370:
2218:Circulant matrices
2212:symmetric matrices
2182:
1933:
1904:
1878:
1849:
1797:
1769:
1749:
1718:
1668:
1648:
1628:
1608:
1588:
1560:
1353:
1307:
1261:
1241:
1221:
1201:
1165:
1116:
1081:
1003:
983:
963:
943:
923:
903:
883:
856:
820:
784:
760:
728:
708:
683:
632:
623:
587:
548:
509:
470:
434:
381:
337:
294:
274:
195:
145:
92:
56:
36:
2177:
2081:
1772:{\displaystyle I}
1721:{\displaystyle A}
1671:{\displaystyle C}
1651:{\displaystyle B}
1631:{\displaystyle C}
1611:{\displaystyle B}
1591:{\displaystyle A}
1264:{\displaystyle B}
1244:{\displaystyle A}
1224:{\displaystyle B}
1204:{\displaystyle A}
1031:of two commuting
1006:{\displaystyle B}
986:{\displaystyle A}
966:{\displaystyle B}
946:{\displaystyle A}
926:{\displaystyle A}
906:{\displaystyle B}
886:{\displaystyle A}
787:{\displaystyle P}
731:{\displaystyle B}
711:{\displaystyle A}
297:{\displaystyle P}
59:{\displaystyle B}
39:{\displaystyle A}
2650:
2627:
2626:
2607:
2601:
2600:
2598:
2597:
2583:
2577:
2576:
2574:
2572:
2561:
2555:
2554:
2522:
2516:
2515:
2503:
2497:
2496:
2494:
2492:
2487:
2479:
2473:
2467:
2465:
2464:
2459:
2454:
2453:
2435:
2434:
2412:
2410:
2409:
2404:
2402:
2401:
2379:
2377:
2376:
2371:
2366:
2365:
2330:
2324:
2323:
2305:
2299:
2298:
2280:
2274:
2273:
2255:
2222:commutative ring
2191:
2189:
2188:
2183:
2178:
2176:
2175:
2174:
2165:
2164:
2139:
2136:
2131:
2115:
2110:
2092:
2091:
2082:
2080:
2076:
2075:
2057:
2056:
2032:
2031:
2010:
2009:
1996:
1982:
1979:
1974:
1942:
1940:
1939:
1934:
1932:
1931:
1926:
1913:
1911:
1910:
1905:
1887:
1885:
1884:
1879:
1858:
1856:
1855:
1850:
1848:
1847:
1842:
1806:
1804:
1803:
1798:
1778:
1776:
1775:
1770:
1758:
1756:
1755:
1750:
1727:
1725:
1724:
1719:
1677:
1675:
1674:
1669:
1657:
1655:
1654:
1649:
1637:
1635:
1634:
1629:
1617:
1615:
1614:
1609:
1597:
1595:
1594:
1589:
1569:
1567:
1566:
1561:
1547:
1546:
1537:
1536:
1524:
1523:
1514:
1513:
1498:
1497:
1488:
1487:
1478:
1477:
1462:
1461:
1452:
1451:
1442:
1441:
1426:
1425:
1416:
1415:
1403:
1402:
1393:
1392:
1362:
1360:
1359:
1354:
1352:
1351:
1342:
1341:
1316:
1314:
1313:
1308:
1306:
1305:
1296:
1295:
1270:
1268:
1267:
1262:
1250:
1248:
1247:
1242:
1230:
1228:
1227:
1222:
1210:
1208:
1207:
1202:
1174:
1172:
1171:
1166:
1161:
1160:
1148:
1147:
1125:
1123:
1122:
1117:
1090:
1088:
1087:
1082:
1080:
1079:
1067:
1066:
1013:'s eigenvectors.
1012:
1010:
1009:
1004:
992:
990:
989:
984:
972:
970:
969:
964:
952:
950:
949:
944:
932:
930:
929:
924:
912:
910:
909:
904:
892:
890:
889:
884:
865:
863:
862:
857:
849:
848:
829:
827:
826:
821:
813:
812:
793:
791:
790:
785:
769:
767:
766:
761:
737:
735:
734:
729:
717:
715:
714:
709:
692:
690:
689:
684:
676:
675:
641:
639:
638:
633:
628:
627:
592:
591:
553:
552:
514:
513:
475:
474:
439:
438:
390:
388:
387:
382:
346:
344:
343:
338:
333:
332:
323:
322:
303:
301:
300:
295:
283:
281:
280:
275:
273:
272:
254:
253:
236:upper triangular
204:
202:
201:
196:
194:
193:
175:
174:
154:
152:
151:
146:
101:
99:
98:
93:
65:
63:
62:
57:
45:
43:
42:
37:
2658:
2657:
2653:
2652:
2651:
2649:
2648:
2647:
2633:
2632:
2631:
2630:
2608:
2604:
2595:
2593:
2585:
2584:
2580:
2570:
2568:
2563:
2562:
2558:
2523:
2519:
2504:
2500:
2490:
2488:
2485:
2481:
2480:
2476:
2443:
2439:
2424:
2420:
2418:
2415:
2414:
2391:
2387:
2385:
2382:
2381:
2355:
2351:
2340:
2337:
2336:
2331:
2327:
2320:
2306:
2302:
2295:
2285:Matrix Analysis
2281:
2277:
2270:
2260:Matrix Analysis
2256:
2252:
2247:
2231:
2198:
2170:
2166:
2154:
2150:
2143:
2138:
2132:
2121:
2111:
2100:
2087:
2083:
2065:
2061:
2052:
2048:
2027:
2023:
2005:
2001:
1997:
1983:
1981:
1975:
1964:
1952:
1949:
1948:
1927:
1922:
1921:
1919:
1916:
1915:
1893:
1890:
1889:
1864:
1861:
1860:
1843:
1838:
1837:
1835:
1832:
1831:
1819: ×
1792:
1789:
1788:
1783: ×
1764:
1761:
1760:
1741:
1738:
1737:
1732: ×
1713:
1710:
1709:
1704: ×
1680:identity matrix
1663:
1660:
1659:
1643:
1640:
1639:
1623:
1620:
1619:
1603:
1600:
1599:
1583:
1580:
1579:
1542:
1538:
1532:
1528:
1519:
1515:
1509:
1505:
1493:
1489:
1483:
1479:
1473:
1469:
1457:
1453:
1447:
1443:
1437:
1433:
1421:
1417:
1411:
1407:
1398:
1394:
1388:
1384:
1370:
1367:
1366:
1347:
1343:
1337:
1333:
1322:
1319:
1318:
1301:
1297:
1291:
1287:
1276:
1273:
1272:
1256:
1253:
1252:
1236:
1233:
1232:
1216:
1213:
1212:
1196:
1193:
1192:
1156:
1152:
1143:
1139:
1131:
1128:
1127:
1096:
1093:
1092:
1075:
1071:
1062:
1058:
1056:
1053:
1052:
998:
995:
994:
978:
975:
974:
958:
955:
954:
938:
935:
934:
918:
915:
914:
898:
895:
894:
878:
875:
874:
841:
837:
835:
832:
831:
805:
801:
799:
796:
795:
794:such that both
779:
776:
775:
743:
740:
739:
723:
720:
719:
703:
700:
699:
671:
667:
653:
650:
649:
622:
621:
616:
610:
609:
604:
594:
593:
586:
585:
580:
574:
573:
568:
558:
557:
547:
546:
541:
535:
534:
529:
519:
518:
508:
507:
502:
496:
495:
490:
480:
479:
469:
468:
463:
457:
456:
451:
441:
440:
433:
432:
427:
421:
420:
415:
405:
404:
402:
399:
398:
352:
349:
348:
328:
324:
315:
311:
309:
306:
305:
289:
286:
285:
268:
264:
249:
245:
243:
240:
239:
215:
189:
185:
170:
166:
164:
161:
160:
110:
107:
106:
75:
72:
71:
51:
48:
47:
31:
28:
27:
17:
12:
11:
5:
2656:
2646:
2645:
2629:
2628:
2618:(1): 222–231,
2602:
2578:
2556:
2517:
2498:
2474:
2470:block diagonal
2457:
2452:
2449:
2446:
2442:
2438:
2433:
2430:
2427:
2423:
2400:
2397:
2394:
2390:
2369:
2364:
2361:
2358:
2354:
2350:
2347:
2344:
2325:
2318:
2300:
2293:
2275:
2268:
2249:
2248:
2246:
2243:
2230:
2227:
2226:
2225:
2215:
2208:
2202:
2197:
2194:
2193:
2192:
2181:
2173:
2169:
2163:
2160:
2157:
2153:
2149:
2146:
2142:
2135:
2130:
2127:
2124:
2120:
2114:
2109:
2106:
2103:
2099:
2095:
2090:
2086:
2079:
2074:
2071:
2068:
2064:
2060:
2055:
2051:
2047:
2044:
2041:
2038:
2035:
2030:
2026:
2022:
2019:
2016:
2013:
2008:
2004:
2000:
1995:
1992:
1989:
1986:
1978:
1973:
1970:
1967:
1963:
1959:
1956:
1930:
1925:
1914:matrices over
1903:
1900:
1897:
1877:
1874:
1871:
1868:
1846:
1841:
1828:
1796:
1768:
1748:
1745:
1717:
1698:
1691:representation
1684:
1667:
1647:
1627:
1607:
1587:
1572:
1571:
1570:
1559:
1556:
1553:
1550:
1545:
1541:
1535:
1531:
1527:
1522:
1518:
1512:
1508:
1504:
1501:
1496:
1492:
1486:
1482:
1476:
1472:
1468:
1465:
1460:
1456:
1450:
1446:
1440:
1436:
1432:
1429:
1424:
1420:
1414:
1410:
1406:
1401:
1397:
1391:
1387:
1383:
1380:
1377:
1374:
1350:
1346:
1340:
1336:
1332:
1329:
1326:
1304:
1300:
1294:
1290:
1286:
1283:
1280:
1260:
1240:
1220:
1200:
1180:
1164:
1159:
1155:
1151:
1146:
1142:
1138:
1135:
1115:
1112:
1109:
1106:
1103:
1100:
1078:
1074:
1070:
1065:
1061:
1025:
1014:
1002:
982:
973:commute, then
962:
942:
922:
902:
882:
871:
855:
852:
847:
844:
840:
819:
816:
811:
808:
804:
783:
770:) if they are
759:
756:
753:
750:
747:
727:
707:
695:
694:
682:
679:
674:
670:
666:
663:
660:
657:
645:
644:
643:
642:
631:
626:
620:
617:
615:
612:
611:
608:
605:
603:
600:
599:
597:
590:
584:
581:
579:
576:
575:
572:
569:
567:
564:
563:
561:
556:
551:
545:
542:
540:
537:
536:
533:
530:
528:
525:
524:
522:
517:
512:
506:
503:
501:
498:
497:
494:
491:
489:
486:
485:
483:
478:
473:
467:
464:
462:
459:
458:
455:
452:
450:
447:
446:
444:
437:
431:
428:
426:
423:
422:
419:
416:
414:
411:
410:
408:
380:
377:
374:
371:
368:
365:
362:
359:
356:
336:
331:
327:
321:
318:
314:
293:
271:
267:
263:
260:
257:
252:
248:
214:
211:
192:
188:
184:
181:
178:
173:
169:
144:
141:
138:
135:
132:
129:
126:
123:
120:
117:
114:
91:
88:
85:
82:
79:
55:
35:
21:linear algebra
15:
9:
6:
4:
3:
2:
2655:
2644:
2643:Matrix theory
2641:
2640:
2638:
2625:
2621:
2617:
2613:
2606:
2592:
2588:
2582:
2566:
2560:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2521:
2513:
2509:
2502:
2484:
2478:
2471:
2455:
2450:
2447:
2444:
2440:
2436:
2431:
2428:
2425:
2421:
2398:
2395:
2392:
2388:
2362:
2359:
2356:
2352:
2345:
2342:
2334:
2329:
2321:
2319:9780521839402
2315:
2311:
2304:
2296:
2294:9780521839402
2290:
2286:
2279:
2271:
2269:9780521839402
2265:
2261:
2254:
2250:
2242:
2240:
2236:
2223:
2219:
2216:
2213:
2209:
2206:
2205:Jordan blocks
2203:
2200:
2199:
2179:
2171:
2167:
2161:
2158:
2155:
2151:
2147:
2144:
2140:
2128:
2125:
2122:
2118:
2107:
2104:
2101:
2097:
2093:
2088:
2084:
2072:
2069:
2066:
2062:
2058:
2053:
2049:
2042:
2036:
2033:
2028:
2024:
2014:
2011:
2006:
2002:
1990:
1984:
1971:
1968:
1965:
1961:
1957:
1954:
1946:
1928:
1901:
1898:
1895:
1872:
1866:
1844:
1829:
1826:
1822:
1818:
1814:
1810:
1794:
1786:
1782:
1766:
1746:
1743:
1735:
1731:
1715:
1707:
1703:
1699:
1696:
1692:
1688:
1687:Lie's theorem
1685:
1683:eigenvectors.
1681:
1665:
1645:
1625:
1605:
1585:
1577:
1573:
1557:
1554:
1551:
1548:
1543:
1539:
1533:
1525:
1520:
1516:
1510:
1502:
1499:
1494:
1490:
1484:
1474:
1466:
1463:
1458:
1454:
1448:
1438:
1430:
1427:
1422:
1418:
1412:
1404:
1399:
1395:
1389:
1381:
1378:
1375:
1372:
1365:
1364:
1348:
1344:
1338:
1330:
1327:
1324:
1302:
1298:
1292:
1284:
1281:
1278:
1258:
1238:
1218:
1198:
1189:
1185:
1181:
1178:
1157:
1153:
1149:
1144:
1140:
1133:
1110:
1107:
1104:
1098:
1076:
1072:
1063:
1059:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1023:
1019:
1015:
1000:
980:
960:
940:
920:
900:
880:
872:
869:
853:
850:
845:
842:
838:
817:
814:
809:
806:
802:
781:
773:
757:
754:
751:
748:
745:
725:
705:
697:
696:
680:
677:
672:
664:
661:
658:
647:
646:
629:
624:
618:
613:
606:
601:
595:
588:
582:
577:
570:
565:
559:
554:
549:
543:
538:
531:
526:
520:
515:
510:
504:
499:
492:
487:
481:
476:
471:
465:
460:
453:
448:
442:
435:
429:
424:
417:
412:
406:
397:
396:
394:
375:
372:
369:
366:
363:
357:
354:
334:
329:
325:
319:
316:
312:
291:
269:
265:
261:
258:
255:
250:
246:
237:
233:
229:
225:
221:
217:
216:
210:
208:
190:
186:
182:
179:
176:
171:
167:
158:
142:
139:
136:
133:
130:
127:
121:
118:
115:
105:
89:
86:
83:
80:
77:
69:
53:
33:
26:
22:
2615:
2611:
2605:
2594:. Retrieved
2590:
2581:
2569:. Retrieved
2559:
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2520:
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2507:
2501:
2489:. Retrieved
2477:
2328:
2309:
2303:
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2278:
2259:
2253:
2232:
1820:
1816:
1784:
1780:
1733:
1729:
1705:
1701:
1638:, and still
1578:: A matrix
1040:
1036:
1022:simple roots
206:
159:of matrices
67:
66:are said to
18:
1188:eigenspaces
1043:with their
1029:eigenvalues
220:eigenspaces
205:is said to
155:is zero. A
2596:2022-07-10
2245:References
1576:transitive
304:such that
104:commutator
2571:August 4,
2551:0012-7094
2241:in 1878.
2239:Frobenius
2159:−
2148:−
2134:∞
2119:∏
2113:∞
2098:∏
2070:−
2059:−
2043:⋯
2034:−
2012:−
1977:∞
1962:∑
1899:×
1795:λ
1744:λ
1544:†
1530:Λ
1521:†
1507:Λ
1495:†
1481:Λ
1471:Λ
1459:†
1445:Λ
1435:Λ
1423:†
1409:Λ
1400:†
1386:Λ
1349:†
1335:Λ
1303:†
1289:Λ
1184:Hermitian
1177:Frobenius
1154:β
1141:α
1073:β
1069:↔
1060:α
1049:multisets
1035:matrices
843:−
807:−
738:commute (
516:≠
370:…
358:∈
317:−
259:…
180:…
137:−
2637:Category
2196:Examples
1825:subgroup
1759:, where
868:diagonal
393:converse
25:matrices
2514:: 1–63.
2491:10 July
2229:History
1945:W. Feit
1811:of the
1779:is the
1708:matrix
1033:complex
207:commute
68:commute
2549:
2316:
2291:
2266:
2235:Cayley
1859:, let
1809:center
391:. The
23:, two
2614:, 3,
2537:(1).
2486:(PDF)
1813:group
1693:of a
1047:(the
232:bases
2573:2018
2547:ISSN
2493:2022
2314:ISBN
2289:ISBN
2264:ISBN
1658:and
1618:and
1317:and
1251:and
1211:and
1182:Two
953:and
893:and
866:are
830:and
718:and
226:are
46:and
2620:doi
2539:doi
1815:of
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873:If
157:set
70:if
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2639::
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1558:.
1555:A
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1549:=
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1534:1
1526:U
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1511:2
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1475:2
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1464:=
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1428:=
1419:U
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1379:=
1376:B
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846:1
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116:A
113:[
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