Knowledge (XXG)

Matrix unit

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1679: 1712: 183: 330: 95: 664: 543: 360: 1337: 255: 82: 1753: 1551: 770: 1642: 1561: 1327: 260: 1746: 717: 418:
When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix
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generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.
1601: 1530: 1412: 1272: 869: 756: 1471: 1054: 859: 217: 1417: 1154: 1004: 999: 834: 809: 804: 1678: 1611: 969: 799: 779: 190: 43: 549: 428: 1777: 1632: 1606: 1184: 989: 979: 335: 1683: 1637: 1627: 1581: 1576: 1505: 1441: 1307: 1044: 1039: 974: 964: 829: 732: 178:{\displaystyle E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}} 234: 1694: 1481: 1476: 1466: 1446: 1407: 1402: 1231: 1226: 1211: 1206: 1197: 1192: 1139: 1034: 984: 929: 899: 894: 874: 864: 824: 28: 1723: 1689: 1657: 1586: 1525: 1520: 1500: 1436: 1342: 1312: 1297: 1282: 1277: 1216: 1169: 1144: 1134: 1105: 1024: 1019: 994: 924: 904: 814: 794: 57: 1387: 1322: 1302: 1287: 1267: 1251: 1149: 1080: 1070: 1029: 914: 884: 363: 1719: 1647: 1591: 1571: 1556: 1515: 1392: 1352: 1317: 1241: 1180: 1159: 1100: 1090: 1075: 1009: 954: 944: 939: 849: 35: 24: 1766: 1652: 1510: 1451: 1382: 1372: 1367: 1292: 1221: 1095: 1085: 1014: 934: 919: 854: 705: 370: 1535: 1492: 1397: 1110: 1049: 959: 839: 1711: 1377: 1347: 1115: 819: 412: 385: 20: 1428: 889: 1662: 1236: 46:
with only one nonzero entry with value 1. The matrix unit with a 1 in the
1596: 415:(induced by the same two vector norms) of a matrix unit is equal to 1. 731:
Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm".
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The product of two matrix units of the same square shape
1727: 574: 453: 117: 552: 431: 338: 263: 237: 98: 60: 1620: 1544: 1490: 1426: 1260: 1178: 1124: 1063: 787: 658: 537: 354: 324: 249: 177: 76: 325:{\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},} 1747: 764: 8: 84:. For example, the 3 by 3 matrix unit with 1754: 1740: 1338:Fundamental (linear differential equation) 771: 757: 749: 736: 639: 615: 591: 573: 560: 551: 501: 489: 477: 452: 436: 430: 343: 337: 310: 297: 281: 268: 262: 236: 112: 103: 97: 65: 59: 1643:Matrix representation of conic sections 675: 7: 1708: 1706: 708:(1999). "Chapter 17: Matrix Rings". 700: 698: 696: 694: 1726:. You can help Knowledge (XXG) by 14: 1710: 1677: 1545:Used in science and engineering 718:Springer Science+Business Media 788:Explicitly constrained entries 659:{\displaystyle AE_{23}=\left.} 538:{\displaystyle E_{23}A=\left.} 1: 1562:Fundamental (computer vision) 714:Graduate Texts in Mathematics 710:Lectures on Modules and Rings 355:{\displaystyle \delta _{jk}} 1328:Duplication and elimination 1127:eigenvalues or eigenvectors 687:. Prentice Hall. p. 9. 396:matrix units in the set of 1799: 1705: 1261:With specific applications 890:Discrete Fourier Transform 18: 1671: 1552:Cabibbo–Kobayashi–Maskawa 1179:Satisfying conditions on 250:{\displaystyle n\times n} 54:th column is denoted as 19:Not to be confused with 910:Generalized permutation 257:satisfies the relation 1722:-related article is a 1684:Mathematics portal 660: 539: 356: 326: 251: 179: 78: 77:{\displaystyle E_{ij}} 16:Concept in mathematics 661: 540: 380:matrices over a ring 357: 327: 252: 180: 79: 1783:Linear algebra stubs 550: 429: 336: 261: 235: 191:standard unit vector 96: 58: 1633:Linear independence 880:Diagonally dominant 720:. pp. 461–479. 198:single-entry matrix 1638:Matrix exponential 1628:Jordan normal form 1462:Fisher information 1333:Euclidean distance 1247:Totally unimodular 656: 647: 535: 526: 352: 322: 247: 216:matrix units is a 175: 169: 74: 1735: 1734: 1703: 1702: 1695:Category:Matrices 1567:Fuzzy associative 1457:Doubly stochastic 1165:Positive-definite 845:Block tridiagonal 716:. Vol. 189. 388:of the subset of 29:invertible matrix 1790: 1756: 1749: 1742: 1714: 1707: 1690:List of matrices 1682: 1681: 1658:Row echelon form 1602:State transition 1531:Seidel adjacency 1413:Totally positive 1273:Alternating sign 870:Complex Hadamard 773: 766: 759: 750: 743: 742: 740: 728: 722: 721: 702: 689: 688: 683:Artin, Michael. 680: 665: 663: 662: 657: 652: 648: 644: 643: 620: 619: 596: 595: 565: 564: 544: 542: 541: 536: 531: 527: 506: 505: 494: 493: 482: 481: 441: 440: 361: 359: 358: 353: 351: 350: 331: 329: 328: 323: 318: 317: 305: 304: 289: 288: 276: 275: 256: 254: 253: 248: 220:of the space of 184: 182: 181: 176: 174: 173: 108: 107: 83: 81: 80: 75: 73: 72: 1798: 1797: 1793: 1792: 1791: 1789: 1788: 1787: 1773:Sparse matrices 1763: 1762: 1761: 1760: 1704: 1699: 1676: 1667: 1616: 1540: 1486: 1422: 1256: 1174: 1120: 1059: 860:Centrosymmetric 783: 777: 747: 746: 730: 729: 725: 704: 703: 692: 682: 681: 677: 672: 646: 645: 635: 633: 628: 622: 621: 611: 609: 604: 598: 597: 587: 585: 580: 569: 556: 548: 547: 525: 524: 519: 514: 508: 507: 497: 495: 485: 483: 473: 470: 469: 464: 459: 448: 432: 427: 426: 364:Kronecker delta 339: 334: 333: 306: 293: 277: 264: 259: 258: 233: 232: 206: 168: 167: 162: 157: 151: 150: 145: 140: 134: 133: 128: 123: 113: 99: 94: 93: 61: 56: 55: 32: 17: 12: 11: 5: 1796: 1794: 1786: 1785: 1780: 1775: 1765: 1764: 1759: 1758: 1751: 1744: 1736: 1733: 1732: 1720:linear algebra 1715: 1701: 1700: 1698: 1697: 1692: 1687: 1672: 1669: 1668: 1666: 1665: 1660: 1655: 1650: 1648:Perfect matrix 1645: 1640: 1635: 1630: 1624: 1622: 1618: 1617: 1615: 1614: 1609: 1604: 1599: 1594: 1589: 1584: 1579: 1574: 1569: 1564: 1559: 1554: 1548: 1546: 1542: 1541: 1539: 1538: 1533: 1528: 1523: 1518: 1513: 1508: 1503: 1497: 1495: 1488: 1487: 1485: 1484: 1479: 1474: 1469: 1464: 1459: 1454: 1449: 1444: 1439: 1433: 1431: 1424: 1423: 1421: 1420: 1418:Transformation 1415: 1410: 1405: 1400: 1395: 1390: 1385: 1380: 1375: 1370: 1365: 1360: 1355: 1350: 1345: 1340: 1335: 1330: 1325: 1320: 1315: 1310: 1305: 1300: 1295: 1290: 1285: 1280: 1275: 1270: 1264: 1262: 1258: 1257: 1255: 1254: 1249: 1244: 1239: 1234: 1229: 1224: 1219: 1214: 1209: 1204: 1195: 1189: 1187: 1176: 1175: 1173: 1172: 1167: 1162: 1157: 1155:Diagonalizable 1152: 1147: 1142: 1137: 1131: 1129: 1125:Conditions on 1122: 1121: 1119: 1118: 1113: 1108: 1103: 1098: 1093: 1088: 1083: 1078: 1073: 1067: 1065: 1061: 1060: 1058: 1057: 1052: 1047: 1042: 1037: 1032: 1027: 1022: 1017: 1012: 1007: 1005:Skew-symmetric 1002: 1000:Skew-Hermitian 997: 992: 987: 982: 977: 972: 967: 962: 957: 952: 947: 942: 937: 932: 927: 922: 917: 912: 907: 902: 897: 892: 887: 882: 877: 872: 867: 862: 857: 852: 847: 842: 837: 835:Block-diagonal 832: 827: 822: 817: 812: 810:Anti-symmetric 807: 805:Anti-Hermitian 802: 797: 791: 789: 785: 784: 778: 776: 775: 768: 761: 753: 745: 744: 723: 706:Lam, Tsit-Yuen 690: 674: 673: 671: 668: 667: 666: 655: 651: 642: 638: 634: 632: 629: 627: 624: 623: 618: 614: 610: 608: 605: 603: 600: 599: 594: 590: 586: 584: 581: 579: 576: 575: 572: 568: 563: 559: 555: 545: 534: 530: 523: 520: 518: 515: 513: 510: 509: 504: 500: 496: 492: 488: 484: 480: 476: 472: 471: 468: 465: 463: 460: 458: 455: 454: 451: 447: 444: 439: 435: 404:matrices over 349: 346: 342: 321: 316: 313: 309: 303: 300: 296: 292: 287: 284: 280: 274: 271: 267: 246: 243: 240: 205: 202: 172: 166: 163: 161: 158: 156: 153: 152: 149: 146: 144: 141: 139: 136: 135: 132: 129: 127: 124: 122: 119: 118: 116: 111: 106: 102: 71: 68: 64: 36:linear algebra 25:unitary matrix 15: 13: 10: 9: 6: 4: 3: 2: 1795: 1784: 1781: 1779: 1776: 1774: 1771: 1770: 1768: 1757: 1752: 1750: 1745: 1743: 1738: 1737: 1731: 1729: 1725: 1721: 1716: 1713: 1709: 1696: 1693: 1691: 1688: 1686: 1685: 1680: 1674: 1673: 1670: 1664: 1661: 1659: 1656: 1654: 1653:Pseudoinverse 1651: 1649: 1646: 1644: 1641: 1639: 1636: 1634: 1631: 1629: 1626: 1625: 1623: 1621:Related terms 1619: 1613: 1612:Z (chemistry) 1610: 1608: 1605: 1603: 1600: 1598: 1595: 1593: 1590: 1588: 1585: 1583: 1580: 1578: 1575: 1573: 1570: 1568: 1565: 1563: 1560: 1558: 1555: 1553: 1550: 1549: 1547: 1543: 1537: 1534: 1532: 1529: 1527: 1524: 1522: 1519: 1517: 1514: 1512: 1509: 1507: 1504: 1502: 1499: 1498: 1496: 1494: 1489: 1483: 1480: 1478: 1475: 1473: 1470: 1468: 1465: 1463: 1460: 1458: 1455: 1453: 1450: 1448: 1445: 1443: 1440: 1438: 1435: 1434: 1432: 1430: 1425: 1419: 1416: 1414: 1411: 1409: 1406: 1404: 1401: 1399: 1396: 1394: 1391: 1389: 1386: 1384: 1381: 1379: 1376: 1374: 1371: 1369: 1366: 1364: 1361: 1359: 1356: 1354: 1351: 1349: 1346: 1344: 1341: 1339: 1336: 1334: 1331: 1329: 1326: 1324: 1321: 1319: 1316: 1314: 1311: 1309: 1306: 1304: 1301: 1299: 1296: 1294: 1291: 1289: 1286: 1284: 1281: 1279: 1276: 1274: 1271: 1269: 1266: 1265: 1263: 1259: 1253: 1250: 1248: 1245: 1243: 1240: 1238: 1235: 1233: 1230: 1228: 1225: 1223: 1220: 1218: 1215: 1213: 1210: 1208: 1205: 1203: 1199: 1196: 1194: 1191: 1190: 1188: 1186: 1182: 1177: 1171: 1168: 1166: 1163: 1161: 1158: 1156: 1153: 1151: 1148: 1146: 1143: 1141: 1138: 1136: 1133: 1132: 1130: 1128: 1123: 1117: 1114: 1112: 1109: 1107: 1104: 1102: 1099: 1097: 1094: 1092: 1089: 1087: 1084: 1082: 1079: 1077: 1074: 1072: 1069: 1068: 1066: 1062: 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: 973: 971: 970:Pentadiagonal 968: 966: 963: 961: 958: 956: 953: 951: 948: 946: 943: 941: 938: 936: 933: 931: 928: 926: 923: 921: 918: 916: 913: 911: 908: 906: 903: 901: 898: 896: 893: 891: 888: 886: 883: 881: 878: 876: 873: 871: 868: 866: 863: 861: 858: 856: 853: 851: 848: 846: 843: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 816: 813: 811: 808: 806: 803: 801: 800:Anti-diagonal 798: 796: 793: 792: 790: 786: 781: 774: 769: 767: 762: 760: 755: 754: 751: 739: 734: 727: 724: 719: 715: 711: 707: 701: 699: 697: 695: 691: 686: 679: 676: 669: 653: 649: 640: 636: 630: 625: 616: 612: 606: 601: 592: 588: 582: 577: 570: 566: 561: 557: 553: 546: 532: 528: 521: 516: 511: 502: 498: 490: 486: 478: 474: 466: 461: 456: 449: 445: 442: 437: 433: 425: 424: 423: 421: 416: 414: 409: 407: 403: 399: 395: 391: 387: 383: 379: 375: 372: 369:The group of 367: 365: 347: 344: 340: 319: 314: 311: 307: 301: 298: 294: 290: 285: 282: 278: 272: 269: 265: 244: 241: 238: 229: 227: 223: 219: 215: 211: 203: 201: 199: 194: 192: 188: 170: 164: 159: 154: 147: 142: 137: 130: 125: 120: 114: 109: 104: 100: 91: 87: 69: 66: 62: 53: 49: 45: 41: 37: 30: 26: 22: 1728:expanding it 1717: 1675: 1607:Substitution 1493:graph theory 990:Quaternionic 980:Persymmetric 949: 726: 709: 684: 678: 419: 417: 410: 405: 401: 397: 393: 389: 381: 377: 373: 368: 230: 225: 221: 213: 209: 207: 197: 195: 186: 89: 85: 51: 47: 39: 33: 1582:Hamiltonian 1506:Biadjacency 1442:Correlation 1358:Householder 1308:Commutation 1045:Vandermonde 1040:Tridiagonal 975:Permutation 965:Nonnegative 950:Matrix unit 830:Bisymmetric 413:matrix norm 386:centralizer 208:The set of 187:vector unit 50:th row and 40:matrix unit 21:unit matrix 1778:1 (number) 1767:Categories 1482:Transition 1477:Stochastic 1447:Covariance 1429:statistics 1408:Symplectic 1403:Similarity 1232:Unimodular 1227:Orthogonal 1212:Involutory 1207:Invertible 1202:Projection 1198:Idempotent 1140:Convergent 1035:Triangular 985:Polynomial 930:Hessenberg 900:Equivalent 895:Elementary 875:Copositive 865:Conference 825:Bidiagonal 670:References 228:matrices. 204:Properties 1663:Wronskian 1587:Irregular 1577:Gell-Mann 1526:Laplacian 1521:Incidence 1501:Adjacency 1472:Precision 1437:Centering 1343:Generator 1313:Confusion 1298:Circulant 1278:Augmented 1237:Unipotent 1217:Nilpotent 1193:Congruent 1170:Stieltjes 1145:Defective 1135:Companion 1106:Redheffer 1025:Symmetric 1020:Sylvester 995:Signature 925:Hermitian 905:Frobenius 815:Arrowhead 795:Alternant 738:0908.2741 341:δ 295:δ 242:× 1491:Used in 1427:Used in 1388:Rotation 1363:Jacobian 1323:Distance 1303:Cofactor 1288:Carleman 1268:Adjugate 1252:Weighing 1185:inverses 1181:products 1150:Definite 1081:Identity 1071:Exchange 1064:Constant 1030:Toeplitz 915:Hadamard 885:Diagonal 88:= 1 and 1592:Overlap 1557:Density 1516:Edmonds 1393:Seifert 1353:Hessian 1318:Coxeter 1242:Unitary 1160:Hurwitz 1091:Of ones 1076:Hilbert 1010:Skyline 955:Metzler 945:Logical 940:Integer 850:Boolean 782:classes 685:Algebra 384:is the 362:is the 92:= 2 is 1511:Degree 1452:Design 1383:Random 1373:Payoff 1368:Moment 1293:Cartan 1283:BĂ©zout 1222:Normal 1096:Pascal 1086:Lehmer 1015:Sparse 935:Hollow 920:Hankel 855:Cauchy 780:Matrix 371:scalar 332:where 44:matrix 1718:This 1572:Gamma 1536:Tutte 1398:Shear 1111:Shift 1101:Pauli 1050:Walsh 960:Moore 840:Block 733:arXiv 218:basis 189:is a 42:is a 27:, or 1724:stub 1378:Pick 1348:Gram 1116:Zero 820:Band 411:The 400:-by- 392:-by- 376:-by- 38:, a 1467:Hat 1200:or 1183:or 224:by 212:by 34:In 1769:: 712:. 693:^ 641:32 617:22 593:12 562:23 503:33 491:32 479:31 438:23 422:: 408:. 366:. 196:A 193:. 185:A 105:12 23:, 1755:e 1748:t 1741:v 1730:. 1597:S 1055:Z 772:e 765:t 758:v 741:. 735:: 654:. 650:] 637:a 631:0 626:0 613:a 607:0 602:0 589:a 583:0 578:0 571:[ 567:= 558:E 554:A 533:. 529:] 522:0 517:0 512:0 499:a 487:a 475:a 467:0 462:0 457:0 450:[ 446:= 443:A 434:E 420:A 406:R 402:n 398:n 394:n 390:n 382:R 378:n 374:n 348:k 345:j 320:, 315:l 312:i 308:E 302:k 299:j 291:= 286:l 283:k 279:E 273:j 270:i 266:E 245:n 239:n 226:n 222:m 214:n 210:m 171:] 165:0 160:0 155:0 148:0 143:0 138:0 131:0 126:1 121:0 115:[ 110:= 101:E 90:j 86:i 70:j 67:i 63:E 52:j 48:i 31:.

Index

unit matrix
unitary matrix
invertible matrix
linear algebra
matrix
standard unit vector
basis
Kronecker delta
scalar
centralizer
matrix norm




Lam, Tsit-Yuen
Graduate Texts in Mathematics
Springer Science+Business Media
arXiv
0908.2741
v
t
e
Matrix
Alternant
Anti-diagonal
Anti-Hermitian
Anti-symmetric
Arrowhead
Band

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