3620:
1978:
532:
1365:
1834:
390:
1228:
838:
1973:{\displaystyle L_{ij}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&\ddots &&&\\&&m&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}}
527:{\displaystyle T_{i,j}={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&0&&1&&\\&&&\ddots &&&\\&&1&&0&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}}
1360:{\displaystyle D_{i}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&m&&&\\&&&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}}
2229:
573:
1564:
1184:
317:
1655:
233:
2329:
155:
2464:
1790:
1037:
910:
2391:
1717:
967:
833:{\displaystyle _{k,l}={\begin{cases}0&k\neq i,k\neq j,k\neq l\\1&k\neq i,k\neq j,k=l\\0&k=i,l\neq j\\1&k=i,l=j\\0&k=j,l\neq i\\1&k=j,l=i\\\end{cases}}}
2069:
3278:
1422:
1041:
For theoretical considerations, the row-switching transformation can be obtained from row-addition and row-multiplication transformations introduced below because
3492:
2711:
3583:
1044:
2624:
2591:
2571:
3502:
3268:
2681:
247:
1578:
173:
2642:
2553:
1214:(usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the
108:
There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):
3303:
2850:
2581:
93:
3067:
2704:
3142:
3298:
2820:
2492:
3402:
3273:
3187:
2254:
120:
3507:
3397:
3105:
2785:
2400:
1726:
97:
976:
3656:
3542:
3471:
3353:
3213:
2810:
2697:
852:
3412:
2995:
2800:
3358:
3095:
2945:
2940:
2775:
2750:
2745:
2342:
2224:{\displaystyle _{k,l}={\begin{cases}0&k\neq l,k\neq i,l\neq j\\1&k=l\\m&k=i,l=j\end{cases}}}
3619:
1668:
921:
3552:
2910:
2740:
2720:
2497:
2482:
1211:
85:
47:
39:
2125:
1472:
620:
3573:
3547:
3125:
2930:
2920:
2469:
70:
3624:
3578:
3568:
3522:
3517:
3446:
3382:
3248:
2985:
2980:
2915:
2905:
2770:
355:
1559:{\displaystyle _{k,l}={\begin{cases}0&k\neq l\\1&k=l,k\neq i\\m&k=l,k=i\end{cases}}}
326:
is an elementary matrix, as described below, to apply the elementary row operation to a matrix
3635:
3422:
3417:
3407:
3387:
3348:
3343:
3172:
3167:
3152:
3147:
3138:
3133:
3080:
2975:
2925:
2870:
2840:
2815:
2805:
2765:
2677:
2638:
2620:
2606:
2587:
2567:
2549:
2334:
3630:
3598:
3527:
3466:
3461:
3441:
3377:
3283:
3253:
3238:
3223:
3218:
3157:
3110:
3085:
3075:
3046:
2965:
2960:
2935:
2865:
2845:
2755:
2735:
2612:
2507:
2502:
338:. The elementary matrix for any row operation is obtained by executing the operation on the
89:
3328:
3263:
3243:
3228:
3208:
3192:
3090:
3021:
3011:
2970:
2855:
2825:
1660:
381:
339:
43:
3588:
3532:
3512:
3497:
3456:
3333:
3293:
3258:
3182:
3121:
3100:
3041:
3031:
3016:
2950:
2895:
2885:
2880:
2790:
2669:
2537:
2487:
2597:
3650:
3593:
3451:
3392:
3323:
3313:
3308:
3233:
3162:
3036:
3026:
2955:
2875:
2860:
2795:
2241:
164:
Each element in a row can be multiplied by a non-zero constant. It is also known as
3476:
3433:
3338:
3051:
2990:
2900:
2780:
343:
3318:
3288:
3056:
2890:
2760:
915:
31:
2616:
3369:
2830:
73:. Left multiplication (pre-multiplication) by an elementary matrix represents
17:
46:
by one single elementary row operation. The elementary matrices generate the
27:
Matrix which differs from the identity matrix by one elementary row operation
3603:
3177:
242:
A row can be replaced by the sum of that row and a multiple of another row.
3537:
1816:. The corresponding elementary matrix is the identity matrix but with an
1179:{\displaystyle T_{i,j}=D_{i}(-1)\,L_{i,j}(-1)\,L_{j,i}(1)\,L_{i,j}(-1).}
312:{\displaystyle R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j}
372:. The corresponding elementary matrix is obtained by swapping row
2689:
1650:{\displaystyle D_{i}(m)^{-1}=D_{i}\left({\tfrac {1}{m}}\right).}
228:{\displaystyle kR_{i}\rightarrow R_{i},\ {\mbox{where }}k\neq 0}
2693:
77:, while right multiplication (post-multiplication) represents
2217:
1552:
826:
2586:, Society for Industrial and Applied Mathematics (SIAM),
115:
A row within the matrix can be switched with another row.
1868:
1629:
1259:
418:
294:
210:
2403:
2345:
2257:
2072:
1837:
1729:
1671:
1581:
1425:
1231:
1047:
979:
924:
855:
576:
393:
250:
176:
123:
342:. This fact can be understood as an instance of the
3561:
3485:
3431:
3367:
3201:
3119:
3065:
3004:
2728:
2458:
2385:
2323:
2223:
1972:
1784:
1711:
1649:
1558:
1359:
1178:
1031:
961:
904:
832:
526:
311:
227:
149:
2653:Elementary Linear Algebra (Applications Version)
2441:
2404:
2346:
1767:
1730:
1672:
1014:
980:
925:
2705:
8:
1800:The final type of row operation on a matrix
360:The first type of row operation on a matrix
2676:(5th ed.), Wellesley-Cambridge Press,
1194:The next type of row operation on a matrix
368:with their counterparts on a different row
3279:Fundamental (linear differential equation)
2712:
2698:
2690:
2583:Matrix Analysis and Applied Linear Algebra
2324:{\displaystyle L_{ij}(m)^{-1}=L_{ij}(-m).}
150:{\displaystyle R_{i}\leftrightarrow R_{j}}
2459:{\displaystyle \det(L_{ij}(m)A)=\det(A).}
2414:
2402:
2356:
2344:
2297:
2281:
2262:
2256:
2120:
2105:
2080:
2071:
1863:
1842:
1836:
1785:{\displaystyle \det(D_{i}(m)A)=m\det(A).}
1740:
1728:
1682:
1670:
1628:
1618:
1602:
1586:
1580:
1467:
1452:
1433:
1424:
1254:
1236:
1230:
1149:
1144:
1123:
1118:
1094:
1089:
1071:
1052:
1046:
990:
978:
935:
923:
887:
871:
860:
854:
615:
600:
584:
575:
545:is the matrix produced by exchanging row
413:
398:
392:
293:
284:
271:
255:
249:
209:
197:
184:
175:
141:
128:
122:
1032:{\displaystyle \det(T_{i,j}A)=-\det(A).}
3584:Matrix representation of conic sections
2526:
2519:
2251:The inverse of this matrix is given by
1575:The inverse of this matrix is given by
969:It follows that for any square matrix
849:The inverse of this matrix is itself:
334:by the elementary matrix on the left,
84:Elementary row operations are used in
2664:(7th ed.), Pearson Prentice Hall
2635:Linear Algebra: A Modern Introduction
2538:Linear algebra § Further reading
905:{\displaystyle T_{i,j}^{-1}=T_{i,j}.}
346:applied to the category of matrices.
7:
2580:Meyer, Carl D. (February 15, 2001),
2468:Row-addition transforms satisfy the
2240:These transformations are a kind of
364:switches all matrix elements on row
2655:(9th ed.), Wiley International
2564:Linear Algebra and Its Applications
2386:{\displaystyle \det(L_{ij}(m))=1.}
25:
2562:Lay, David C. (August 22, 2005),
2548:(2nd ed.), Springer-Verlag,
1712:{\displaystyle \det(D_{i}(m))=m.}
962:{\displaystyle \det(T_{i,j})=-1.}
918:of the identity matrix is unity,
3618:
2662:Linear Algebra With Applications
2566:(3rd ed.), Addison Wesley,
96:to further reduce the matrix to
3486:Used in science and engineering
2393:Therefore, for a square matrix
2333:The matrix and its inverse are
1723:(of the correct size), we have
1719:Therefore, for a square matrix
1659:The matrix and its inverse are
1198:multiplies all elements on row
1190:Row-multiplying transformations
973:(of the correct size), we have
2729:Explicitly constrained entries
2674:Introduction to Linear Algebra
2450:
2444:
2435:
2429:
2423:
2407:
2397:(of the correct size) we have
2374:
2371:
2365:
2349:
2315:
2306:
2278:
2271:
2102:
2098:
2092:
2073:
1857:
1851:
1776:
1770:
1758:
1752:
1746:
1733:
1697:
1694:
1688:
1675:
1599:
1592:
1449:
1445:
1439:
1426:
1248:
1242:
1170:
1161:
1141:
1135:
1115:
1106:
1086:
1077:
1023:
1017:
1005:
983:
947:
928:
597:
577:
277:
190:
134:
1:
3503:Fundamental (computer vision)
2637:(2nd ed.), Brooks/Cole,
2049:Coefficient wise, the matrix
1416:matrix is defined by :
560:Coefficient wise, the matrix
350:Row-switching transformations
2030:is the matrix produced from
2000:is the matrix produced from
1796:Row-addition transformations
1387:is the matrix produced from
79:elementary column operations
3269:Duplication and elimination
3068:eigenvalues or eigenvectors
2544:Axler, Sheldon Jay (1997),
3673:
3202:With specific applications
2831:Discrete Fourier Transform
2617:10.1142/9789811286018_0005
2535:
2493:System of linear equations
353:
3612:
3493:Cabibbo–Kobayashi–Maskawa
3120:Satisfying conditions on
2546:Linear Algebra Done Right
1218:th position, where it is
104:Elementary row operations
92:. They are also used in
75:elementary row operations
2660:Leon, Steven J. (2006),
2608:Starting Category Theory
98:reduced row echelon form
94:Gauss–Jordan elimination
2851:Generalized permutation
2605:Perrone, Paolo (2024),
1808:multiplied by a scalar
42:which differs from the
3625:Mathematics portal
2651:Anton, Howard (2005),
2460:
2387:
2325:
2225:
2063:is defined by :
1974:
1786:
1713:
1651:
1560:
1402:Coefficient wise, the
1361:
1180:
1033:
963:
906:
834:
567:is defined by :
528:
313:
229:
151:
88:to reduce a matrix to
2633:Poole, David (2006),
2461:
2388:
2326:
2226:
1975:
1787:
1714:
1652:
1561:
1362:
1181:
1034:
964:
907:
835:
529:
314:
230:
152:
2611:, World Scientific,
2498:Matrix (mathematics)
2483:Gaussian elimination
2401:
2343:
2255:
2070:
1835:
1727:
1669:
1579:
1423:
1229:
1045:
977:
922:
853:
574:
391:
248:
174:
121:
86:Gaussian elimination
48:general linear group
3574:Linear independence
2821:Diagonally dominant
2470:Steinberg relations
2335:triangular matrices
1391:by multiplying row
879:
3579:Matrix exponential
3569:Jordan normal form
3403:Fisher information
3274:Euclidean distance
3188:Totally unimodular
2529:, pp. 119–120
2456:
2383:
2321:
2244:, also known as a
2221:
2216:
1970:
1964:
1782:
1709:
1647:
1638:
1556:
1551:
1357:
1351:
1176:
1029:
959:
902:
856:
830:
825:
524:
518:
356:Permutation matrix
309:
298:
225:
214:
161:Row multiplication
147:
3644:
3643:
3636:Category:Matrices
3508:Fuzzy associative
3398:Doubly stochastic
3106:Positive-definite
2786:Block tridiagonal
2626:978-981-12-8600-1
2593:978-0-89871-454-8
2573:978-0-321-28713-7
1661:diagonal matrices
1637:
330:, one multiplies
297:
213:
208:
36:elementary matrix
16:(Redirected from
3664:
3631:List of matrices
3623:
3622:
3599:Row echelon form
3543:State transition
3472:Seidel adjacency
3354:Totally positive
3214:Alternating sign
2811:Complex Hadamard
2714:
2707:
2700:
2691:
2686:
2683:978-09802327-7-6
2665:
2656:
2647:
2629:
2601:
2596:, archived from
2576:
2558:
2530:
2524:
2508:Frobenius matrix
2503:LU decomposition
2465:
2463:
2462:
2457:
2422:
2421:
2396:
2392:
2390:
2389:
2384:
2364:
2363:
2330:
2328:
2327:
2322:
2305:
2304:
2289:
2288:
2270:
2269:
2230:
2228:
2227:
2222:
2220:
2219:
2116:
2115:
2091:
2090:
2062:
2045:
2041:
2037:
2033:
2029:
2015:
2011:
2007:
2003:
1999:
1979:
1977:
1976:
1971:
1969:
1968:
1957:
1956:
1955:
1954:
1953:
1952:
1949:
1943:
1942:
1941:
1940:
1939:
1936:
1935:
1929:
1923:
1922:
1919:
1918:
1917:
1911:
1910:
1909:
1906:
1905:
1904:
1903:
1897:
1896:
1893:
1892:
1891:
1890:
1889:
1883:
1880:
1879:
1878:
1877:
1876:
1875:
1850:
1849:
1827:
1819:
1815:
1811:
1807:
1803:
1791:
1789:
1788:
1783:
1745:
1744:
1722:
1718:
1716:
1715:
1710:
1687:
1686:
1656:
1654:
1653:
1648:
1643:
1639:
1630:
1623:
1622:
1610:
1609:
1591:
1590:
1565:
1563:
1562:
1557:
1555:
1554:
1463:
1462:
1438:
1437:
1415:
1398:
1394:
1390:
1386:
1366:
1364:
1363:
1358:
1356:
1355:
1344:
1343:
1342:
1341:
1340:
1339:
1336:
1330:
1329:
1328:
1327:
1326:
1323:
1322:
1316:
1315:
1314:
1313:
1310:
1309:
1308:
1302:
1301:
1300:
1297:
1296:
1295:
1294:
1288:
1287:
1284:
1283:
1282:
1281:
1280:
1274:
1271:
1270:
1269:
1268:
1267:
1266:
1241:
1240:
1221:
1217:
1209:
1205:
1201:
1197:
1185:
1183:
1182:
1177:
1160:
1159:
1134:
1133:
1105:
1104:
1076:
1075:
1063:
1062:
1038:
1036:
1035:
1030:
1001:
1000:
972:
968:
966:
965:
960:
946:
945:
911:
909:
908:
903:
898:
897:
878:
870:
839:
837:
836:
831:
829:
828:
611:
610:
595:
594:
566:
556:
552:
548:
544:
533:
531:
530:
525:
523:
522:
511:
510:
509:
508:
507:
506:
503:
497:
496:
495:
494:
493:
490:
489:
483:
477:
476:
473:
472:
471:
465:
464:
463:
460:
459:
453:
447:
446:
443:
442:
441:
440:
439:
433:
430:
429:
428:
427:
426:
425:
409:
408:
379:
375:
371:
367:
363:
337:
333:
329:
325:
318:
316:
315:
310:
299:
295:
289:
288:
276:
275:
260:
259:
234:
232:
231:
226:
215:
211:
206:
202:
201:
189:
188:
156:
154:
153:
148:
146:
145:
133:
132:
90:row echelon form
68:
62:
21:
3672:
3671:
3667:
3666:
3665:
3663:
3662:
3661:
3647:
3646:
3645:
3640:
3617:
3608:
3557:
3481:
3427:
3363:
3197:
3115:
3061:
3000:
2801:Centrosymmetric
2724:
2718:
2684:
2670:Strang, Gilbert
2668:
2659:
2650:
2645:
2632:
2627:
2604:
2594:
2579:
2574:
2561:
2556:
2543:
2540:
2534:
2533:
2525:
2521:
2516:
2479:
2410:
2399:
2398:
2394:
2352:
2341:
2340:
2293:
2277:
2258:
2253:
2252:
2237:
2215:
2214:
2191:
2185:
2184:
2173:
2167:
2166:
2131:
2121:
2101:
2076:
2068:
2067:
2055:
2050:
2043:
2039:
2035:
2031:
2022:
2017:
2013:
2009:
2005:
2001:
1989:
1984:
1963:
1962:
1950:
1948:
1937:
1934:
1928:
1920:
1916:
1907:
1902:
1894:
1888:
1881:
1874:
1864:
1838:
1833:
1832:
1821:
1817:
1813:
1809:
1805:
1801:
1798:
1736:
1725:
1724:
1720:
1678:
1667:
1666:
1624:
1614:
1598:
1582:
1577:
1576:
1572:
1550:
1549:
1526:
1520:
1519:
1496:
1490:
1489:
1478:
1468:
1448:
1429:
1421:
1420:
1408:
1403:
1396:
1392:
1388:
1376:
1371:
1350:
1349:
1337:
1335:
1324:
1321:
1311:
1307:
1298:
1293:
1285:
1279:
1272:
1265:
1255:
1232:
1227:
1226:
1219:
1215:
1207:
1203:
1199:
1195:
1192:
1145:
1119:
1090:
1067:
1048:
1043:
1042:
986:
975:
974:
970:
931:
920:
919:
883:
851:
850:
846:
824:
823:
800:
794:
793:
770:
764:
763:
740:
734:
733:
710:
704:
703:
668:
662:
661:
626:
616:
596:
580:
572:
571:
565:
561:
554:
550:
546:
542:
538:
517:
516:
504:
502:
491:
488:
482:
474:
470:
461:
458:
452:
444:
438:
431:
424:
414:
394:
389:
388:
382:identity matrix
377:
373:
369:
365:
361:
358:
352:
340:identity matrix
335:
331:
327:
323:
280:
267:
251:
246:
245:
193:
180:
172:
171:
137:
124:
119:
118:
106:
64:
56:
50:
44:identity matrix
28:
23:
22:
15:
12:
11:
5:
3670:
3668:
3660:
3659:
3657:Linear algebra
3649:
3648:
3642:
3641:
3639:
3638:
3633:
3628:
3613:
3610:
3609:
3607:
3606:
3601:
3596:
3591:
3589:Perfect matrix
3586:
3581:
3576:
3571:
3565:
3563:
3559:
3558:
3556:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3489:
3487:
3483:
3482:
3480:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3438:
3436:
3429:
3428:
3426:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3385:
3380:
3374:
3372:
3365:
3364:
3362:
3361:
3359:Transformation
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3221:
3216:
3211:
3205:
3203:
3199:
3198:
3196:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3145:
3136:
3130:
3128:
3117:
3116:
3114:
3113:
3108:
3103:
3098:
3096:Diagonalizable
3093:
3088:
3083:
3078:
3072:
3070:
3066:Conditions on
3063:
3062:
3060:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3014:
3008:
3006:
3002:
3001:
2999:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2948:
2946:Skew-symmetric
2943:
2941:Skew-Hermitian
2938:
2933:
2928:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2776:Block-diagonal
2773:
2768:
2763:
2758:
2753:
2751:Anti-symmetric
2748:
2746:Anti-Hermitian
2743:
2738:
2732:
2730:
2726:
2725:
2719:
2717:
2716:
2709:
2702:
2694:
2688:
2687:
2682:
2666:
2657:
2648:
2643:
2630:
2625:
2602:
2592:
2577:
2572:
2559:
2554:
2532:
2531:
2527:Perrone (2024)
2518:
2517:
2515:
2512:
2511:
2510:
2505:
2500:
2495:
2490:
2488:Linear algebra
2485:
2478:
2475:
2474:
2473:
2466:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2420:
2417:
2413:
2409:
2406:
2382:
2379:
2376:
2373:
2370:
2367:
2362:
2359:
2355:
2351:
2348:
2338:
2331:
2320:
2317:
2314:
2311:
2308:
2303:
2300:
2296:
2292:
2287:
2284:
2280:
2276:
2273:
2268:
2265:
2261:
2249:
2236:
2233:
2232:
2231:
2218:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2190:
2187:
2186:
2183:
2180:
2177:
2174:
2172:
2169:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2130:
2127:
2126:
2124:
2119:
2114:
2111:
2108:
2104:
2100:
2097:
2094:
2089:
2086:
2083:
2079:
2075:
2053:
2020:
1987:
1981:
1980:
1967:
1961:
1958:
1951:
1947:
1944:
1938:
1933:
1930:
1927:
1924:
1921:
1915:
1912:
1908:
1901:
1898:
1895:
1887:
1884:
1882:
1873:
1870:
1869:
1867:
1862:
1859:
1856:
1853:
1848:
1845:
1841:
1797:
1794:
1793:
1792:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1743:
1739:
1735:
1732:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1685:
1681:
1677:
1674:
1664:
1657:
1646:
1642:
1636:
1633:
1627:
1621:
1617:
1613:
1608:
1605:
1601:
1597:
1594:
1589:
1585:
1571:
1568:
1567:
1566:
1553:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1525:
1522:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1495:
1492:
1491:
1488:
1485:
1482:
1479:
1477:
1474:
1473:
1471:
1466:
1461:
1458:
1455:
1451:
1447:
1444:
1441:
1436:
1432:
1428:
1406:
1374:
1368:
1367:
1354:
1348:
1345:
1338:
1334:
1331:
1325:
1320:
1317:
1312:
1306:
1303:
1299:
1292:
1289:
1286:
1278:
1275:
1273:
1264:
1261:
1260:
1258:
1253:
1250:
1247:
1244:
1239:
1235:
1210:is a non-zero
1191:
1188:
1187:
1186:
1175:
1172:
1169:
1166:
1163:
1158:
1155:
1152:
1148:
1143:
1140:
1137:
1132:
1129:
1126:
1122:
1117:
1114:
1111:
1108:
1103:
1100:
1097:
1093:
1088:
1085:
1082:
1079:
1074:
1070:
1066:
1061:
1058:
1055:
1051:
1039:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
999:
996:
993:
989:
985:
982:
958:
955:
952:
949:
944:
941:
938:
934:
930:
927:
912:
901:
896:
893:
890:
886:
882:
877:
874:
869:
866:
863:
859:
845:
842:
841:
840:
827:
822:
819:
816:
813:
810:
807:
804:
801:
799:
796:
795:
792:
789:
786:
783:
780:
777:
774:
771:
769:
766:
765:
762:
759:
756:
753:
750:
747:
744:
741:
739:
736:
735:
732:
729:
726:
723:
720:
717:
714:
711:
709:
706:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
667:
664:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
625:
622:
621:
619:
614:
609:
606:
603:
599:
593:
590:
587:
583:
579:
563:
540:
535:
534:
521:
515:
512:
505:
501:
498:
492:
487:
484:
481:
478:
475:
469:
466:
462:
457:
454:
451:
448:
445:
437:
434:
432:
423:
420:
419:
417:
412:
407:
404:
401:
397:
351:
348:
320:
319:
308:
305:
302:
292:
287:
283:
279:
274:
270:
266:
263:
258:
254:
243:
240:
236:
235:
224:
221:
218:
205:
200:
196:
192:
187:
183:
179:
169:
162:
158:
157:
144:
140:
136:
131:
127:
116:
113:
105:
102:
52:
26:
24:
18:Row operations
14:
13:
10:
9:
6:
4:
3:
2:
3669:
3658:
3655:
3654:
3652:
3637:
3634:
3632:
3629:
3627:
3626:
3621:
3615:
3614:
3611:
3605:
3602:
3600:
3597:
3595:
3594:Pseudoinverse
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3566:
3564:
3562:Related terms
3560:
3554:
3553:Z (chemistry)
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3490:
3488:
3484:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3439:
3437:
3435:
3430:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3375:
3373:
3371:
3366:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3206:
3204:
3200:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3140:
3137:
3135:
3132:
3131:
3129:
3127:
3123:
3118:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3071:
3069:
3064:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3009:
3007:
3003:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2911:Pentadiagonal
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2741:Anti-diagonal
2739:
2737:
2734:
2733:
2731:
2727:
2722:
2715:
2710:
2708:
2703:
2701:
2696:
2695:
2692:
2685:
2679:
2675:
2671:
2667:
2663:
2658:
2654:
2649:
2646:
2644:0-534-99845-3
2640:
2636:
2631:
2628:
2622:
2618:
2614:
2610:
2609:
2603:
2600:on 2009-10-31
2599:
2595:
2589:
2585:
2584:
2578:
2575:
2569:
2565:
2560:
2557:
2555:0-387-98259-0
2551:
2547:
2542:
2541:
2539:
2528:
2523:
2520:
2513:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2480:
2476:
2471:
2467:
2453:
2447:
2438:
2432:
2426:
2418:
2415:
2411:
2380:
2377:
2368:
2360:
2357:
2353:
2339:
2336:
2332:
2318:
2312:
2309:
2301:
2298:
2294:
2290:
2285:
2282:
2274:
2266:
2263:
2259:
2250:
2247:
2246:transvections
2243:
2242:shear mapping
2239:
2238:
2234:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2188:
2181:
2178:
2175:
2170:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2128:
2122:
2117:
2112:
2109:
2106:
2095:
2087:
2084:
2081:
2077:
2066:
2065:
2064:
2060:
2056:
2047:
2038:times column
2027:
2023:
1998:
1994:
1990:
1965:
1959:
1945:
1931:
1925:
1913:
1899:
1885:
1871:
1865:
1860:
1854:
1846:
1843:
1839:
1831:
1830:
1829:
1825:
1795:
1779:
1773:
1764:
1761:
1755:
1749:
1741:
1737:
1706:
1703:
1700:
1691:
1683:
1679:
1665:
1662:
1658:
1644:
1640:
1634:
1631:
1625:
1619:
1615:
1611:
1606:
1603:
1595:
1587:
1583:
1574:
1573:
1569:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1523:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1493:
1486:
1483:
1480:
1475:
1469:
1464:
1459:
1456:
1453:
1442:
1434:
1430:
1419:
1418:
1417:
1413:
1409:
1400:
1385:
1381:
1377:
1352:
1346:
1332:
1318:
1304:
1290:
1276:
1262:
1256:
1251:
1245:
1237:
1233:
1225:
1224:
1223:
1213:
1189:
1173:
1167:
1164:
1156:
1153:
1150:
1146:
1138:
1130:
1127:
1124:
1120:
1112:
1109:
1101:
1098:
1095:
1091:
1083:
1080:
1072:
1068:
1064:
1059:
1056:
1053:
1049:
1040:
1026:
1020:
1011:
1008:
1002:
997:
994:
991:
987:
956:
953:
950:
942:
939:
936:
932:
917:
913:
899:
894:
891:
888:
884:
880:
875:
872:
867:
864:
861:
857:
848:
847:
843:
820:
817:
814:
811:
808:
805:
802:
797:
790:
787:
784:
781:
778:
775:
772:
767:
760:
757:
754:
751:
748:
745:
742:
737:
730:
727:
724:
721:
718:
715:
712:
707:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
665:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
623:
617:
612:
607:
604:
601:
591:
588:
585:
581:
570:
569:
568:
558:
519:
513:
499:
485:
479:
467:
455:
449:
435:
421:
415:
410:
405:
402:
399:
395:
387:
386:
385:
383:
357:
349:
347:
345:
341:
306:
303:
300:
290:
285:
281:
272:
268:
264:
261:
256:
252:
244:
241:
238:
237:
222:
219:
216:
203:
198:
194:
185:
181:
177:
170:
167:
163:
160:
159:
142:
138:
129:
125:
117:
114:
112:Row switching
111:
110:
109:
103:
101:
99:
95:
91:
87:
82:
80:
76:
72:
67:
60:
55:
49:
45:
41:
37:
33:
19:
3616:
3548:Substitution
3434:graph theory
2931:Quaternionic
2921:Persymmetric
2835:
2673:
2661:
2652:
2634:
2607:
2598:the original
2582:
2563:
2545:
2522:
2245:
2058:
2051:
2048:
2025:
2018:
1996:
1992:
1985:
1982:
1823:
1799:
1411:
1404:
1401:
1383:
1379:
1372:
1369:
1193:
559:
536:
359:
344:Yoneda lemma
321:
239:Row addition
165:
107:
83:
78:
74:
65:
58:
53:
35:
29:
3523:Hamiltonian
3447:Biadjacency
3383:Correlation
3299:Householder
3249:Commutation
2986:Vandermonde
2981:Tridiagonal
2916:Permutation
2906:Nonnegative
2891:Matrix unit
2771:Bisymmetric
916:determinant
296:where
212:where
32:mathematics
3423:Transition
3418:Stochastic
3388:Covariance
3370:statistics
3349:Symplectic
3344:Similarity
3173:Unimodular
3168:Orthogonal
3153:Involutory
3148:Invertible
3143:Projection
3139:Idempotent
3081:Convergent
2976:Triangular
2926:Polynomial
2871:Hessenberg
2841:Equivalent
2836:Elementary
2816:Copositive
2806:Conference
2766:Bidiagonal
2536:See also:
2514:References
2235:Properties
2042:to column
2034:by adding
2008:times row
2004:by adding
1828:position.
1570:Properties
914:Since the
844:Properties
354:See also:
3604:Wronskian
3528:Irregular
3518:Gell-Mann
3467:Laplacian
3462:Incidence
3442:Adjacency
3413:Precision
3378:Centering
3284:Generator
3254:Confusion
3239:Circulant
3219:Augmented
3178:Unipotent
3158:Nilpotent
3134:Congruent
3111:Stieltjes
3086:Defective
3076:Companion
3047:Redheffer
2966:Symmetric
2961:Sylvester
2936:Signature
2866:Hermitian
2846:Frobenius
2756:Arrowhead
2736:Alternant
2310:−
2283:−
2161:≠
2149:≠
2137:≠
1946:⋱
1914:⋱
1886:⋱
1804:adds row
1604:−
1514:≠
1484:≠
1333:⋱
1277:⋱
1165:−
1110:−
1081:−
1012:−
954:−
873:−
788:≠
728:≠
686:≠
674:≠
656:≠
644:≠
632:≠
500:⋱
468:⋱
436:⋱
304:≠
278:→
220:≠
191:→
135:↔
3651:Category
3432:Used in
3368:Used in
3329:Rotation
3304:Jacobian
3264:Distance
3244:Cofactor
3229:Carleman
3209:Adjugate
3193:Weighing
3126:inverses
3122:products
3091:Definite
3022:Identity
3012:Exchange
3005:Constant
2971:Toeplitz
2856:Hadamard
2826:Diagonal
2672:(2016),
2477:See also
549:and row
376:and row
3533:Overlap
3498:Density
3457:Edmonds
3334:Seifert
3294:Hessian
3259:Coxeter
3183:Unitary
3101:Hurwitz
3032:Of ones
3017:Hilbert
2951:Skyline
2896:Metzler
2886:Logical
2881:Integer
2791:Boolean
2723:classes
2016:. And
2012:to row
1820:in the
1812:to row
380:of the
166:scaling
3452:Degree
3393:Design
3324:Random
3314:Payoff
3309:Moment
3234:Cartan
3224:BĂ©zout
3163:Normal
3037:Pascal
3027:Lehmer
2956:Sparse
2876:Hollow
2861:Hankel
2796:Cauchy
2721:Matrix
2680:
2641:
2623:
2590:
2570:
2552:
1212:scalar
1206:where
207:
168:a row.
40:matrix
3513:Gamma
3477:Tutte
3339:Shear
3052:Shift
3042:Pauli
2991:Walsh
2901:Moore
2781:Block
71:field
69:is a
63:when
38:is a
34:, an
3319:Pick
3289:Gram
3057:Zero
2761:Band
2678:ISBN
2639:ISBN
2621:ISBN
2588:ISBN
2568:ISBN
2550:ISBN
1824:i, j
3408:Hat
3141:or
3124:or
2613:doi
2442:det
2405:det
2347:det
2054:i,j
2019:A L
1983:So
1768:det
1731:det
1673:det
1395:by
1370:So
1202:by
1015:det
981:det
926:det
564:i,j
553:of
541:i,j
537:So
384:.
322:If
30:In
3653::
2619:,
2381:1.
2046:.
2021:ij
1988:ij
1399:.
1222:.
957:1.
557:.
336:EA
100:.
81:.
51:GL
3538:S
2996:Z
2713:e
2706:t
2699:v
2615::
2472:.
2454:.
2451:)
2448:A
2445:(
2439:=
2436:)
2433:A
2430:)
2427:m
2424:(
2419:j
2416:i
2412:L
2408:(
2395:A
2378:=
2375:)
2372:)
2369:m
2366:(
2361:j
2358:i
2354:L
2350:(
2337:.
2319:.
2316:)
2313:m
2307:(
2302:j
2299:i
2295:L
2291:=
2286:1
2279:)
2275:m
2272:(
2267:j
2264:i
2260:L
2248:.
2212:j
2209:=
2206:l
2203:,
2200:i
2197:=
2194:k
2189:m
2182:l
2179:=
2176:k
2171:1
2164:j
2158:l
2155:,
2152:i
2146:k
2143:,
2140:l
2134:k
2129:0
2123:{
2118:=
2113:l
2110:,
2107:k
2103:]
2099:)
2096:m
2093:(
2088:j
2085:,
2082:i
2078:L
2074:[
2061:)
2059:m
2057:(
2052:L
2044:j
2040:i
2036:m
2032:A
2028:)
2026:m
2024:(
2014:i
2010:j
2006:m
2002:A
1997:A
1995:)
1993:m
1991:(
1986:L
1966:]
1960:1
1932:1
1926:m
1900:1
1872:1
1866:[
1861:=
1858:)
1855:m
1852:(
1847:j
1844:i
1840:L
1826:)
1822:(
1818:m
1814:i
1810:m
1806:j
1802:A
1780:.
1777:)
1774:A
1771:(
1765:m
1762:=
1759:)
1756:A
1753:)
1750:m
1747:(
1742:i
1738:D
1734:(
1721:A
1707:.
1704:m
1701:=
1698:)
1695:)
1692:m
1689:(
1684:i
1680:D
1676:(
1663:.
1645:.
1641:)
1635:m
1632:1
1626:(
1620:i
1616:D
1612:=
1607:1
1600:)
1596:m
1593:(
1588:i
1584:D
1547:i
1544:=
1541:k
1538:,
1535:l
1532:=
1529:k
1524:m
1517:i
1511:k
1508:,
1505:l
1502:=
1499:k
1494:1
1487:l
1481:k
1476:0
1470:{
1465:=
1460:l
1457:,
1454:k
1450:]
1446:)
1443:m
1440:(
1435:i
1431:D
1427:[
1414:)
1412:m
1410:(
1407:i
1405:D
1397:m
1393:i
1389:A
1384:A
1382:)
1380:m
1378:(
1375:i
1373:D
1353:]
1347:1
1319:1
1305:m
1291:1
1263:1
1257:[
1252:=
1249:)
1246:m
1243:(
1238:i
1234:D
1220:m
1216:i
1208:m
1204:m
1200:i
1196:A
1174:.
1171:)
1168:1
1162:(
1157:j
1154:,
1151:i
1147:L
1142:)
1139:1
1136:(
1131:i
1128:,
1125:j
1121:L
1116:)
1113:1
1107:(
1102:j
1099:,
1096:i
1092:L
1087:)
1084:1
1078:(
1073:i
1069:D
1065:=
1060:j
1057:,
1054:i
1050:T
1027:.
1024:)
1021:A
1018:(
1009:=
1006:)
1003:A
998:j
995:,
992:i
988:T
984:(
971:A
951:=
948:)
943:j
940:,
937:i
933:T
929:(
900:.
895:j
892:,
889:i
885:T
881:=
876:1
868:j
865:,
862:i
858:T
821:i
818:=
815:l
812:,
809:j
806:=
803:k
798:1
791:i
785:l
782:,
779:j
776:=
773:k
768:0
761:j
758:=
755:l
752:,
749:i
746:=
743:k
738:1
731:j
725:l
722:,
719:i
716:=
713:k
708:0
701:l
698:=
695:k
692:,
689:j
683:k
680:,
677:i
671:k
666:1
659:l
653:k
650:,
647:j
641:k
638:,
635:i
629:k
624:0
618:{
613:=
608:l
605:,
602:k
598:]
592:j
589:,
586:i
582:T
578:[
562:T
555:A
551:j
547:i
543:A
539:T
520:]
514:1
486:0
480:1
456:1
450:0
422:1
416:[
411:=
406:j
403:,
400:i
396:T
378:j
374:i
370:j
366:i
362:A
332:A
328:A
324:E
307:j
301:i
291:,
286:i
282:R
273:j
269:R
265:k
262:+
257:i
253:R
223:0
217:k
204:,
199:i
195:R
186:i
182:R
178:k
143:j
139:R
130:i
126:R
66:F
61:)
59:F
57:(
54:n
20:)
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