1854:
910:
881:
437:
262:
432:{\displaystyle {\begin{pmatrix}1&0&0&&\cdots &&0\\0&1&0&&\cdots &&0\\0&0&\ddots &&&&0\\\vdots &&&1&&&\vdots \\&&&&0&&\\&&&&&\ddots &\\0&&&\cdots &&&0\end{pmatrix}}}
220:
If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence.
713:
This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class.
167:, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same
722:
Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true. For example these two matrices are equivalent but not similar:
829:
776:
709:
656:
603:
1512:
85:
480:
460:
255:
1726:
945:
1817:
550:
2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:
541:: Matrix equivalent classes are characterized by rank: two same-sided matrixes are matrix equivalent if and only if they have the same rank.
1736:
1502:
1895:
781:
728:
661:
608:
555:
1537:
1084:
1301:
938:
201:
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions
1376:
1532:
1054:
1636:
1507:
1421:
1741:
1631:
1339:
1019:
206:
1776:
1705:
1587:
1447:
1044:
931:
1646:
1229:
1034:
491:
129:
1592:
1329:
1179:
1174:
1009:
984:
979:
1853:
1786:
1144:
974:
954:
195:
28:
1807:
1781:
1359:
1164:
1154:
1858:
1812:
1802:
1756:
1751:
1680:
1616:
1482:
1219:
1214:
1149:
1139:
1004:
914:
909:
885:
880:
48:
1890:
1869:
1656:
1651:
1641:
1621:
1582:
1577:
1406:
1401:
1386:
1381:
1372:
1367:
1314:
1209:
1159:
1104:
1074:
1069:
1049:
1039:
999:
844:
483:
91:
1864:
1832:
1761:
1700:
1695:
1675:
1611:
1517:
1487:
1472:
1457:
1452:
1391:
1344:
1319:
1309:
1280:
1199:
1194:
1169:
1099:
1079:
989:
969:
230:
213:
1562:
1497:
1477:
1462:
1442:
1426:
1324:
1255:
1245:
1204:
1089:
1059:
839:
149:
913: This article incorporates text from this source, which is available under the
884: This article incorporates text from this source, which is available under the
1822:
1766:
1746:
1731:
1690:
1567:
1527:
1492:
1416:
1355:
1334:
1275:
1265:
1250:
1184:
1129:
1119:
1114:
1024:
465:
445:
240:
234:
164:
17:
1884:
1827:
1685:
1626:
1557:
1547:
1542:
1467:
1396:
1270:
1260:
1189:
1109:
1094:
1029:
1710:
1667:
1572:
1285:
1224:
1134:
1014:
168:
1552:
1522:
1290:
1124:
994:
487:
1603:
1064:
118:
1837:
1411:
1771:
900:
871:
205:
The matrices can be transformed into one another by a combination of
923:
824:{\displaystyle {\begin{pmatrix}1&2\\0&3\\\end{pmatrix}}}
771:{\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}}
704:{\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}}
651:{\displaystyle {\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}}
598:{\displaystyle {\begin{pmatrix}0&0\\0&0\\\end{pmatrix}}}
927:
212:
Two matrices are equivalent if and only if they have the same
163:
The notion of equivalence should not be confused with that of
790:
737:
670:
617:
564:
271:
784:
731:
664:
611:
558:
486:, which generalizes this concept on vector spaces to
468:
448:
265:
243:
51:
1795:
1719:
1665:
1601:
1435:
1353:
1299:
1238:
962:
823:
770:
703:
650:
597:
474:
454:
431:
249:
186:, used both for initial vectors and their images.
79:
531:matrix that is all zeroes except that the first
237:for matrices of the equivalence class of rank
939:
8:
1513:Fundamental (linear differential equation)
946:
932:
924:
785:
783:
732:
730:
665:
663:
612:
610:
559:
557:
467:
447:
266:
264:
242:
128:under two different choices of a pair of
117:. Equivalent matrices represent the same
62:
50:
1818:Matrix representation of conic sections
855:
198:on the space of rectangular matrices.
7:
865:
863:
861:
859:
207:elementary row and column operations
14:
876:(4th ed.). pp. 270–272.
178:under two different choices of a
1852:
908:
879:
482:. This is a special case of the
1720:Used in science and engineering
963:Explicitly constrained entries
462:s on the diagonal is equal to
1:
1737:Fundamental (computer vision)
233:property yields an intuitive
905:(4th ed.). p. 405.
537:diagonal entries are ones.
519:is matrix equivalent to the
1503:Duplication and elimination
1302:eigenvalues or eigenvectors
1912:
1436:With specific applications
1065:Discrete Fourier Transform
80:{\displaystyle B=Q^{-1}AP}
1896:Equivalence (mathematics)
1846:
1727:Cabibbo–Kobayashi–Maskawa
1354:Satisfying conditions on
194:Matrix equivalence is an
1085:Generalized permutation
492:principal ideal domains
1859:Mathematics portal
825:
772:
705:
652:
599:
543:
476:
456:
433:
251:
81:
826:
773:
706:
653:
600:
496:
477:
457:
434:
252:
119:linear transformation
82:
782:
729:
662:
609:
556:
466:
446:
442:where the number of
263:
241:
196:equivalence relation
105:and some invertible
49:
1808:Linear independence
1055:Diagonally dominant
1813:Matrix exponential
1803:Jordan normal form
1637:Fisher information
1508:Euclidean distance
1422:Totally unimodular
821:
815:
768:
762:
701:
695:
648:
642:
595:
589:
472:
452:
429:
423:
247:
77:
20:, two rectangular
1878:
1877:
1870:Category:Matrices
1742:Fuzzy associative
1632:Doubly stochastic
1340:Positive-definite
1020:Block tridiagonal
845:Matrix congruence
718:Matrix similarity
484:Smith normal form
475:{\displaystyle k}
455:{\displaystyle 1}
250:{\displaystyle k}
1903:
1865:List of matrices
1857:
1856:
1833:Row echelon form
1777:State transition
1706:Seidel adjacency
1588:Totally positive
1448:Alternating sign
1045:Complex Hadamard
948:
941:
934:
925:
919:
912:
906:
896:
890:
883:
877:
867:
830:
828:
827:
822:
820:
819:
777:
775:
774:
769:
767:
766:
710:
708:
707:
702:
700:
699:
657:
655:
654:
649:
647:
646:
604:
602:
601:
596:
594:
593:
481:
479:
478:
473:
461:
459:
458:
453:
438:
436:
435:
430:
428:
427:
416:
415:
409:
408:
400:
394:
393:
392:
391:
390:
387:
386:
380:
379:
378:
377:
369:
368:
362:
361:
348:
347:
346:
323:
317:
294:
288:
256:
254:
253:
248:
86:
84:
83:
78:
70:
69:
1911:
1910:
1906:
1905:
1904:
1902:
1901:
1900:
1881:
1880:
1879:
1874:
1851:
1842:
1791:
1715:
1661:
1597:
1431:
1349:
1295:
1234:
1035:Centrosymmetric
958:
952:
922:
899:Hefferon, Jim.
898:
897:
893:
870:Hefferon, Jim.
869:
868:
857:
853:
840:Row equivalence
836:
814:
813:
808:
802:
801:
796:
786:
780:
779:
761:
760:
755:
749:
748:
743:
733:
727:
726:
720:
694:
693:
688:
682:
681:
676:
666:
660:
659:
641:
640:
635:
629:
628:
623:
613:
607:
606:
588:
587:
582:
576:
575:
570:
560:
554:
553:
548:
513:matrix of rank
464:
463:
444:
443:
422:
421:
414:
407:
401:
399:
388:
385:
375:
374:
367:
360:
354:
353:
345:
340:
335:
329:
328:
322:
316:
311:
306:
300:
299:
293:
287:
282:
277:
267:
261:
260:
239:
238:
227:
192:
150:change of basis
58:
47:
46:
12:
11:
5:
1909:
1907:
1899:
1898:
1893:
1883:
1882:
1876:
1875:
1873:
1872:
1867:
1862:
1847:
1844:
1843:
1841:
1840:
1835:
1830:
1825:
1823:Perfect matrix
1820:
1815:
1810:
1805:
1799:
1797:
1793:
1792:
1790:
1789:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1744:
1739:
1734:
1729:
1723:
1721:
1717:
1716:
1714:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1678:
1672:
1670:
1663:
1662:
1660:
1659:
1654:
1649:
1644:
1639:
1634:
1629:
1624:
1619:
1614:
1608:
1606:
1599:
1598:
1596:
1595:
1593:Transformation
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1439:
1437:
1433:
1432:
1430:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1384:
1379:
1370:
1364:
1362:
1351:
1350:
1348:
1347:
1342:
1337:
1332:
1330:Diagonalizable
1327:
1322:
1317:
1312:
1306:
1304:
1300:Conditions on
1297:
1296:
1294:
1293:
1288:
1283:
1278:
1273:
1268:
1263:
1258:
1253:
1248:
1242:
1240:
1236:
1235:
1233:
1232:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1180:Skew-symmetric
1177:
1175:Skew-Hermitian
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1010:Block-diagonal
1007:
1002:
997:
992:
987:
985:Anti-symmetric
982:
980:Anti-Hermitian
977:
972:
966:
964:
960:
959:
953:
951:
950:
943:
936:
928:
921:
920:
902:Linear Algebra
891:
873:Linear Algebra
854:
852:
849:
848:
847:
842:
835:
832:
818:
812:
809:
807:
804:
803:
800:
797:
795:
792:
791:
789:
765:
759:
756:
754:
751:
750:
747:
744:
742:
739:
738:
736:
719:
716:
698:
692:
689:
687:
684:
683:
680:
677:
675:
672:
671:
669:
645:
639:
636:
634:
631:
630:
627:
624:
622:
619:
618:
616:
592:
586:
583:
581:
578:
577:
574:
571:
569:
566:
565:
563:
547:
544:
471:
451:
426:
420:
417:
413:
410:
406:
403:
402:
398:
395:
389:
384:
381:
376:
373:
370:
366:
363:
359:
356:
355:
352:
349:
344:
341:
339:
336:
334:
331:
330:
327:
324:
321:
318:
315:
312:
310:
307:
305:
302:
301:
298:
295:
292:
289:
286:
283:
281:
278:
276:
273:
272:
270:
246:
235:canonical form
226:
225:Canonical form
223:
218:
217:
210:
191:
188:
160:respectively.
88:
87:
76:
73:
68:
65:
61:
57:
54:
18:linear algebra
13:
10:
9:
6:
4:
3:
2:
1908:
1897:
1894:
1892:
1889:
1888:
1886:
1871:
1868:
1866:
1863:
1861:
1860:
1855:
1849:
1848:
1845:
1839:
1836:
1834:
1831:
1829:
1828:Pseudoinverse
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1798:
1796:Related terms
1794:
1788:
1787:Z (chemistry)
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1724:
1722:
1718:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1673:
1671:
1669:
1664:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1609:
1607:
1605:
1600:
1594:
1591:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1440:
1438:
1434:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1385:
1383:
1380:
1378:
1374:
1371:
1369:
1366:
1365:
1363:
1361:
1357:
1352:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1307:
1305:
1303:
1298:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1243:
1241:
1237:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1145:Pentadiagonal
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
975:Anti-diagonal
973:
971:
968:
967:
965:
961:
956:
949:
944:
942:
937:
935:
930:
929:
926:
918:
916:
911:
904:
903:
895:
892:
889:
887:
882:
875:
874:
866:
864:
862:
860:
856:
850:
846:
843:
841:
838:
837:
833:
831:
816:
810:
805:
798:
793:
787:
763:
757:
752:
745:
740:
734:
724:
717:
715:
711:
696:
690:
685:
678:
673:
667:
643:
637:
632:
625:
620:
614:
590:
584:
579:
572:
567:
561:
551:
545:
542:
540:
536:
535:
530:
529:
524:
523:
518:
517:
512:
511:
506:
505:
500:
495:
493:
489:
485:
469:
449:
440:
424:
418:
411:
404:
396:
382:
371:
364:
357:
350:
342:
337:
332:
325:
319:
313:
308:
303:
296:
290:
284:
279:
274:
268:
258:
244:
236:
232:
224:
222:
215:
211:
208:
204:
203:
202:
199:
197:
189:
187:
185:
181:
177:
174: →
173:
170:
166:
161:
159:
155:
151:
147:
143:
139:
135:
131:
127:
124: →
123:
120:
116:
112:
108:
104:
100:
96:
93:
74:
71:
66:
63:
59:
55:
52:
45:
44:
43:
41:
37:
33:
30:
27:
23:
19:
1850:
1782:Substitution
1668:graph theory
1165:Quaternionic
1155:Persymmetric
915:CC BY-SA 3.0
907:
901:
894:
886:CC BY-SA 3.0
878:
872:
725:
721:
712:
552:
549:
546:2x2 matrices
538:
533:
532:
527:
526:
521:
520:
515:
514:
509:
508:
503:
502:
498:
497:
488:free modules
441:
259:
228:
219:
200:
193:
183:
179:
175:
171:
169:endomorphism
162:
157:
153:
152:matrices in
145:
141:
137:
133:
125:
121:
114:
110:
106:
102:
98:
94:
89:
39:
35:
31:
25:
21:
15:
1757:Hamiltonian
1681:Biadjacency
1617:Correlation
1533:Householder
1483:Commutation
1220:Vandermonde
1215:Tridiagonal
1150:Permutation
1140:Nonnegative
1125:Matrix unit
1005:Bisymmetric
38:are called
1885:Categories
1657:Transition
1652:Stochastic
1622:Covariance
1604:statistics
1583:Symplectic
1578:Similarity
1407:Unimodular
1402:Orthogonal
1387:Involutory
1382:Invertible
1377:Projection
1373:Idempotent
1315:Convergent
1210:Triangular
1160:Polynomial
1105:Hessenberg
1075:Equivalent
1070:Elementary
1050:Copositive
1040:Conference
1000:Bidiagonal
851:References
190:Properties
165:similarity
148:being the
92:invertible
40:equivalent
1838:Wronskian
1762:Irregular
1752:Gell-Mann
1701:Laplacian
1696:Incidence
1676:Adjacency
1647:Precision
1612:Centering
1518:Generator
1488:Confusion
1473:Circulant
1453:Augmented
1412:Unipotent
1392:Nilpotent
1368:Congruent
1345:Stieltjes
1320:Defective
1310:Companion
1281:Redheffer
1200:Symmetric
1195:Sylvester
1170:Signature
1100:Hermitian
1080:Frobenius
990:Arrowhead
970:Alternant
539:Corollary
412:⋯
397:⋱
372:⋮
358:⋮
343:⋱
320:⋯
291:⋯
182:basis of
90:for some
64:−
1891:Matrices
1666:Used in
1602:Used in
1563:Rotation
1538:Jacobian
1498:Distance
1478:Cofactor
1463:Carleman
1443:Adjugate
1427:Weighing
1360:inverses
1356:products
1325:Definite
1256:Identity
1246:Exchange
1239:Constant
1205:Toeplitz
1090:Hadamard
1060:Diagonal
917:license.
888:license.
834:See also
494:. Thus:
29:matrices
1767:Overlap
1732:Density
1691:Edmonds
1568:Seifert
1528:Hessian
1493:Coxeter
1417:Unitary
1335:Hurwitz
1266:Of ones
1251:Hilbert
1185:Skyline
1130:Metzler
1120:Logical
1115:Integer
1025:Boolean
957:classes
499:Theorem
140:, with
113:matrix
101:matrix
1686:Degree
1627:Design
1558:Random
1548:Payoff
1543:Moment
1468:Cartan
1458:BĂ©zout
1397:Normal
1271:Pascal
1261:Lehmer
1190:Sparse
1110:Hollow
1095:Hankel
1030:Cauchy
955:Matrix
501:: Any
180:single
1747:Gamma
1711:Tutte
1573:Shear
1286:Shift
1276:Pauli
1225:Walsh
1135:Moore
1015:Block
490:over
130:bases
1553:Pick
1523:Gram
1291:Zero
995:Band
231:rank
229:The
214:rank
156:and
144:and
136:and
109:-by-
97:-by-
34:and
24:-by-
1642:Hat
1375:or
1358:or
778:,
658:,
605:,
439:,
257:as
132:of
42:if
16:In
1887::
858:^
1772:S
1230:Z
947:e
940:t
933:v
817:)
811:3
806:0
799:2
794:1
788:(
764:)
758:1
753:0
746:0
741:1
735:(
697:)
691:1
686:0
679:0
674:1
668:(
644:)
638:0
633:0
626:0
621:1
615:(
591:)
585:0
580:0
573:0
568:0
562:(
534:k
528:n
525:x
522:m
516:k
510:n
507:x
504:m
470:k
450:1
425:)
419:0
405:0
383:0
365:1
351:0
338:0
333:0
326:0
314:0
309:1
304:0
297:0
285:0
280:0
275:1
269:(
245:k
216:.
209:.
184:V
176:V
172:V
158:W
154:V
146:Q
142:P
138:W
134:V
126:W
122:V
115:Q
111:m
107:m
103:P
99:n
95:n
75:P
72:A
67:1
60:Q
56:=
53:B
36:B
32:A
26:n
22:m
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
