2855:
2662:
326:
1899:
2850:{\displaystyle {\begin{bmatrix}I_{n}&X\\0&I_{m}\end{bmatrix}}{\begin{bmatrix}A&C\\0&B\end{bmatrix}}{\begin{bmatrix}I_{n}&-X\\0&I_{m}\end{bmatrix}}={\begin{bmatrix}A&0\\0&B\end{bmatrix}}.}
2615:
2562:
2337:
The theorem remains true for real matrices with the caveat that one considers their complex eigenvalues. The proof for the "if" part is still applicable; for the "only if" part, note that both
1376:
773:
732:
653:
612:
2860:
Roth's removal rule does not generalize to infinite-dimensional bounded operators on a Banach space. Nevertheless, Roth's removal rule generalizes to the systems of
Sylvester equations.
2205:
2963:
2423:
2379:
234:
1123:
1405:
1489:
473:
425:
377:
2300:
1202:
565:
1738:
526:
1613:
1068:
77:
3417:
3344:
2461:
2247:
1764:
1149:
941:
860:
691:
1297:
242:
3289:
2332:
1961:
1521:
500:
1547:
2051:
2008:
1700:
1593:
1448:
1007:
883:
816:
2910:
2890:
2131:
2111:
2091:
2071:
2028:
1985:
1919:
1677:
1657:
1633:
1570:
1425:
1262:
1242:
1222:
1027:
984:
961:
903:
793:
445:
397:
349:
106:. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally,
3585:
3451:
3114:
1769:
3046:
However, rewriting the equation in this form is not advised for the numerical solution since this version is costly to solve and can be
2982:
function in that language. In some specific image processing applications, the derived
Sylvester equation has a closed form solution.
2567:
2514:
3512:
3194:
3510:
Wei, Q.; Dobigeon, N.; Tourneret, J.-Y. (2015). "Fast Fusion of Multi-Band Images Based on
Solving a Sylvester Equation".
3192:
Wei, Q.; Dobigeon, N.; Tourneret, J.-Y. (2015). "Fast Fusion of Multi-Band Images Based on
Solving a Sylvester Equation".
3488:
1310:
737:
696:
617:
576:
3165:
2136:
186:
3582:
3449:
Dmytryshyn, Andrii; KÄgström, Bo (2015). "Coupled
Sylvester-type Matrix Equations and Block Diagonalization".
3112:
Dmytryshyn, Andrii; KÄgström, Bo (2015). "Coupled
Sylvester-type Matrix Equations and Block Diagonalization".
1921:
is a nontrivial solution to the aforesaid homogeneous equation, justifying the "only if" part of the theorem.
2869:
2928:
2384:
2340:
219:
2208:
1300:
214:
2997:
1073:
3484:"Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum"
1381:
3465:
3128:
1453:
1928:
1304:
450:
402:
354:
3594:
2252:
1154:
3619:
3421:
538:
1964:
1705:
3460:
3293:
3123:
505:
321:{\displaystyle (I_{m}\otimes A+B^{T}\otimes I_{n})\operatorname {vec} X=\operatorname {vec} C,}
83:
1598:
1032:
41:
3384:
3311:
2428:
2214:
1743:
1128:
908:
827:
658:
3599:
1267:
3531:
3262:
3213:
3151:
3060:
Gerrish, F; Ward, A.G.B (Nov 1998). "Sylvester's matrix equation and Roth's removal rule".
2305:
1934:
1494:
478:
29:
8:
2913:
1526:
3535:
3217:
2033:
1990:
1682:
1575:
1430:
989:
865:
798:
3555:
3521:
3437:
3369:
3237:
3203:
3085:
3077:
2895:
2875:
2617:. The answer is that these two matrices are similar exactly when there exists a matrix
2116:
2096:
2076:
2056:
2013:
1970:
1904:
1662:
1642:
1618:
1555:
1410:
1247:
1227:
1207:
1012:
969:
946:
888:
885:
unknowns and the same number of equations. Hence it is uniquely solvable for any given
778:
430:
382:
334:
3614:
3547:
3441:
3229:
3089:
2991:
2921:
2508:
210:
3373:
3539:
3496:
3470:
3429:
3359:
3221:
3133:
3069:
170:
3559:
3241:
2868:
A classical algorithm for the numerical solution of the
Sylvester equation is the
3589:
3047:
528:
2925:
103:
21:
3501:
3483:
3433:
3381:
Bhatia, R.; Rosenthal, P. (1997). "How and why to solve the operator equation
102:
that obey this equation. All matrices are assumed to have coefficients in the
3608:
3543:
3225:
1894:{\displaystyle AX+XB=A(uv^{*})-(uv^{*})(-B)=\lambda uv^{*}-\lambda uv^{*}=0.}
532:
198:
3551:
3233:
2917:
174:
3364:
3307:
1636:
17:
3081:
2975:
146:
3474:
3137:
3350:
3073:
3526:
3208:
32:
2499:, then one can ask when the following two square matrices of size
3016:
This equation is also commonly written in the equivalent form of
2967:
1029:
be a solution to the abovementioned homogeneous equation. Then
177:. In this case, the condition for the uniqueness of a solution
2610:{\displaystyle {\begin{bmatrix}A&0\\0&B\end{bmatrix}}}
2557:{\displaystyle {\begin{bmatrix}A&C\\0&B\end{bmatrix}}}
204:
1963:
in part (i) of the proof can also be demonstrated by the
2637:
is a solution to a
Sylvester equation. This is known as
2920:, and then solving the resulting triangular system via
2813:
2757:
2721:
2671:
2576:
2523:
1549:
as desired. This proves the "if" part of the theorem.
3387:
3314:
3265:
2931:
2898:
2878:
2665:
2570:
2517:
2431:
2387:
2343:
2308:
2255:
2217:
2139:
2119:
2099:
2079:
2059:
2036:
2016:
1993:
1973:
1937:
1907:
1772:
1746:
1708:
1685:
1665:
1645:
1621:
1601:
1578:
1558:
1529:
1497:
1456:
1433:
1413:
1384:
1313:
1270:
1250:
1230:
1210:
1157:
1131:
1076:
1035:
1015:
992:
972:
949:
911:
891:
868:
830:
801:
781:
740:
699:
661:
620:
579:
541:
508:
481:
453:
433:
405:
385:
357:
337:
245:
222:
44:
3595:
Mathematica function to solve the
Sylvester equation
3509:
3191:
3166:"Functions of a Matrix (GNU Octave (version 4.4.1))"
3259:Sylvester, J. (1884). "Sur l'equations en matrices
2966:arithmetical operations, is used, among others, by
181:is almost the same: There exists a unique solution
3448:
3411:
3338:
3283:
3182:command is deprecated since GNU Octave Version 4.0
3111:
2957:
2904:
2884:
2849:
2609:
2556:
2455:
2417:
2373:
2326:
2294:
2241:
2199:
2125:
2105:
2085:
2065:
2045:
2022:
2002:
1979:
1955:
1913:
1893:
1758:
1732:
1694:
1671:
1651:
1627:
1607:
1587:
1564:
1541:
1515:
1483:
1442:
1419:
1399:
1370:
1291:
1256:
1236:
1216:
1196:
1143:
1117:
1062:
1021:
1001:
978:
955:
935:
897:
877:
854:
810:
787:
767:
726:
685:
647:
606:
559:
520:
494:
467:
439:
419:
391:
371:
343:
320:
236:, we can rewrite Sylvester's equation in the form
228:
71:
461:
457:
413:
409:
365:
361:
3606:
3566:
3452:SIAM Journal on Matrix Analysis and Applications
3115:SIAM Journal on Matrix Analysis and Applications
3600:MATLAB function to solve the Sylvester equation
3380:
141:A Sylvester equation has a unique solution for
98:, the problem is to find the possible matrices
3305:
2924:. This algorithm, whose computational cost is
1371:{\displaystyle \sigma (p(-B))=p(\sigma (-B)),}
531:. In this form, the equation can be seen as a
768:{\displaystyle C\in \mathbb {C} ^{n\times m}}
727:{\displaystyle X\in \mathbb {C} ^{n\times m}}
648:{\displaystyle B\in \mathbb {C} ^{m\times m}}
607:{\displaystyle A\in \mathbb {C} ^{n\times n}}
3583:Online solver for arbitrary sized matrices.
3059:
2093:are coprime. Hence there exist polynomials
2994:, a special case of the Sylvester equation
2463:, and they cannot be zero simultaneously.
3525:
3500:
3464:
3363:
3258:
3207:
3127:
2200:{\displaystyle p(z)f(z)+q(z)g(z)\equiv 1}
1151:by mathematical induction. Consequently,
905:if and only if the homogeneous equation
749:
708:
629:
588:
205:Existence and uniqueness of the solutions
1679:be a corresponding left eigenvector for
1407:denotes the spectrum of a matrix. Since
82:It is named after English mathematician
3306:Bartels, R. H.; Stewart, G. W. (1972).
3607:
2863:
2466:
169:has been considered as an equation of
3513:IEEE Transactions on Image Processing
3481:
2958:{\displaystyle {\mathcal {O}}(n^{3})}
2418:{\displaystyle \mathrm {Im} (uv^{*})}
2374:{\displaystyle \mathrm {Re} (uv^{*})}
173:on a (possibly infinite-dimensional)
2644:One easily checks one direction: If
1987:be the characteristic polynomial of
1244:be the characteristic polynomial of
229:{\displaystyle \operatorname {vec} }
13:
2934:
2471:Given two square complex matrices
2392:
2389:
2348:
2345:
14:
3631:
3576:
3308:"Solution of the matrix equation
2872:, which consists of transforming
1491:does not contain zero, and hence
1009:do not share any eigenvalue. Let
943:admits only the trivial solution
145:exactly when there are no common
114:must be square matrices of sizes
2425:satisfy the homogenous equation
1118:{\displaystyle A^{k}X=X(-B)^{k}}
3571:. Macmillan. pp. 213, 299.
3185:
1400:{\displaystyle \sigma (\cdot )}
157:. More generally, the equation
3482:Lee, S.-G.; Vu, Q.-P. (2011).
3172:
3158:
3144:
3105:
3096:
3053:
3040:
3031:
3010:
2952:
2939:
2412:
2396:
2368:
2352:
2321:
2312:
2283:
2274:
2268:
2259:
2230:
2221:
2188:
2182:
2176:
2170:
2161:
2155:
2149:
2143:
1967:for coprime polynomials. Let
1950:
1941:
1844:
1835:
1832:
1816:
1810:
1794:
1510:
1501:
1484:{\displaystyle p(\sigma (-B))}
1478:
1475:
1466:
1460:
1394:
1388:
1362:
1359:
1350:
1344:
1335:
1332:
1323:
1317:
1280:
1274:
1191:
1182:
1167:
1161:
1106:
1096:
1057:
1048:
291:
246:
1:
3252:
2053:do not share any eigenvalue,
1450:do not share any eigenvalue,
818:do not share any eigenvalue.
468:{\displaystyle n\!\times \!m}
420:{\displaystyle m\!\times \!m}
372:{\displaystyle n\!\times \!n}
2295:{\displaystyle p(-B)f(-B)=I}
1197:{\displaystyle p(A)X=Xp(-B)}
7:
2985:
560:{\displaystyle mn\times mn}
10:
3636:
3569:A survey of Modern Algebra
3152:"Function Reference: Lyap"
3037:Bhatia and Rosenthal, 1997
2998:Algebraic Riccati equation
1733:{\displaystyle X=u{v}^{*}}
1070:, which can be lifted to
3502:10.1016/j.laa.2010.09.034
3434:10.1112/S0024609396001828
3102:Bhatia and Rosenthal, p.3
2870:BartelsâStewart algorithm
1927:As an alternative to the
1635:be a corresponding right
655:, the Sylvester equation
521:{\displaystyle k\times k}
3544:10.1109/TIP.2015.2458572
3226:10.1109/TIP.2015.2458572
3062:The Mathematical Gazette
3003:
1931:, the nonsingularity of
1929:spectral mapping theorem
1608:{\displaystyle \lambda }
1305:spectral mapping theorem
1063:{\displaystyle AX=X(-B)}
862:is a linear system with
72:{\displaystyle AX+XB=C.}
3422:Bull. London Math. Soc.
3412:{\displaystyle AX-XB=Y}
3339:{\displaystyle AX+XB=C}
2456:{\displaystyle AX+XB=0}
2242:{\displaystyle q(-B)=0}
2209:CayleyâHamilton theorem
1759:{\displaystyle X\neq 0}
1301:CayleyâHamilton theorem
1144:{\displaystyle k\geq 0}
936:{\displaystyle AX+XB=0}
855:{\displaystyle AX+XB=C}
686:{\displaystyle AX+XB=C}
122:respectively, and then
3567:Birkhoff and MacLane.
3413:
3340:
3294:C. R. Acad. Sci. Paris
3285:
2959:
2906:
2886:
2851:
2611:
2558:
2457:
2419:
2375:
2328:
2296:
2243:
2201:
2127:
2107:
2087:
2067:
2047:
2024:
2004:
1981:
1957:
1915:
1895:
1760:
1734:
1696:
1673:
1653:
1629:
1609:
1589:
1566:
1543:
1517:
1485:
1444:
1421:
1401:
1372:
1293:
1292:{\displaystyle p(A)=0}
1258:
1238:
1218:
1198:
1145:
1119:
1064:
1023:
1003:
980:
957:
937:
899:
879:
856:
812:
789:
769:
728:
693:has a unique solution
687:
649:
608:
561:
522:
496:
469:
441:
421:
393:
373:
345:
322:
230:
215:vectorization operator
86:. Then given matrices
84:James Joseph Sylvester
73:
3414:
3365:10.1145/361573.361582
3341:
3286:
3284:{\displaystyle px=xq}
2960:
2907:
2887:
2852:
2612:
2559:
2458:
2420:
2376:
2329:
2327:{\displaystyle p(-B)}
2297:
2244:
2202:
2128:
2108:
2088:
2068:
2048:
2025:
2005:
1982:
1958:
1956:{\displaystyle p(-B)}
1916:
1896:
1761:
1735:
1697:
1674:
1654:
1630:
1610:
1590:
1567:
1552:(ii) Now assume that
1544:
1523:is nonsingular. Thus
1518:
1516:{\displaystyle p(-B)}
1486:
1445:
1422:
1402:
1373:
1294:
1259:
1239:
1224:. In particular, let
1219:
1199:
1146:
1120:
1065:
1024:
1004:
981:
958:
938:
900:
880:
857:
813:
790:
770:
729:
688:
650:
609:
562:
523:
497:
495:{\displaystyle I_{k}}
470:
442:
422:
394:
374:
346:
323:
231:
74:
3489:Linear Algebra Appl.
3385:
3312:
3301:(2): 67â71, 115â116.
3263:
2929:
2896:
2876:
2663:
2568:
2515:
2429:
2385:
2341:
2306:
2253:
2215:
2137:
2117:
2097:
2077:
2057:
2034:
2014:
1991:
1971:
1935:
1905:
1770:
1744:
1706:
1683:
1663:
1643:
1619:
1599:
1595:share an eigenvalue
1576:
1556:
1527:
1495:
1454:
1431:
1411:
1382:
1311:
1268:
1248:
1228:
1208:
1155:
1129:
1074:
1033:
1013:
990:
970:
947:
909:
889:
866:
828:
799:
779:
738:
697:
659:
618:
577:
539:
506:
479:
451:
431:
403:
383:
355:
335:
243:
220:
42:
3536:2015ITIP...24.4109W
3218:2015ITIP...24.4109W
2864:Numerical solutions
2639:Roth's removal rule
2467:Roth's removal rule
1542:{\displaystyle X=0}
1204:for any polynomial
3588:2013-07-09 at the
3409:
3336:
3281:
2955:
2902:
2882:
2847:
2838:
2799:
2746:
2710:
2633:. In other words,
2607:
2601:
2554:
2548:
2453:
2415:
2371:
2324:
2292:
2239:
2197:
2123:
2103:
2083:
2063:
2046:{\displaystyle -B}
2043:
2020:
2003:{\displaystyle -B}
2000:
1977:
1953:
1911:
1891:
1756:
1730:
1695:{\displaystyle -B}
1692:
1669:
1649:
1625:
1605:
1588:{\displaystyle -B}
1585:
1562:
1539:
1513:
1481:
1443:{\displaystyle -B}
1440:
1417:
1397:
1368:
1289:
1254:
1234:
1214:
1194:
1141:
1115:
1060:
1019:
1002:{\displaystyle -B}
999:
976:
953:
933:
895:
878:{\displaystyle mn}
875:
852:
811:{\displaystyle -B}
808:
785:
765:
724:
683:
645:
604:
557:
518:
492:
465:
437:
417:
389:
369:
341:
318:
226:
69:
26:Sylvester equation
20:, in the field of
3520:(11): 4109â4121.
3475:10.1137/151005907
3202:(11): 4109â4121.
3138:10.1137/151005907
2992:Lyapunov equation
2922:back-substitution
2905:{\displaystyle B}
2885:{\displaystyle A}
2126:{\displaystyle g}
2106:{\displaystyle f}
2086:{\displaystyle q}
2066:{\displaystyle p}
2023:{\displaystyle A}
1980:{\displaystyle q}
1965:BĂ©zout's identity
1914:{\displaystyle X}
1672:{\displaystyle v}
1652:{\displaystyle A}
1628:{\displaystyle u}
1565:{\displaystyle A}
1420:{\displaystyle A}
1303:; meanwhile, the
1257:{\displaystyle A}
1237:{\displaystyle p}
1217:{\displaystyle p}
1022:{\displaystyle X}
979:{\displaystyle A}
956:{\displaystyle 0}
898:{\displaystyle C}
788:{\displaystyle A}
440:{\displaystyle X}
392:{\displaystyle B}
344:{\displaystyle A}
213:notation and the
211:Kronecker product
185:exactly when the
171:bounded operators
3627:
3572:
3563:
3529:
3506:
3504:
3495:(9): 2097â2109.
3478:
3468:
3445:
3418:
3416:
3415:
3410:
3377:
3367:
3345:
3343:
3342:
3337:
3302:
3290:
3288:
3287:
3282:
3246:
3245:
3211:
3189:
3183:
3181:
3176:
3170:
3169:
3162:
3156:
3155:
3148:
3142:
3141:
3131:
3109:
3103:
3100:
3094:
3093:
3068:(495): 423â430.
3057:
3051:
3044:
3038:
3035:
3029:
3014:
2981:
2973:
2964:
2962:
2961:
2956:
2951:
2950:
2938:
2937:
2911:
2909:
2908:
2903:
2891:
2889:
2888:
2883:
2856:
2854:
2853:
2848:
2843:
2842:
2804:
2803:
2796:
2795:
2769:
2768:
2751:
2750:
2715:
2714:
2707:
2706:
2683:
2682:
2616:
2614:
2613:
2608:
2606:
2605:
2563:
2561:
2560:
2555:
2553:
2552:
2462:
2460:
2459:
2454:
2424:
2422:
2421:
2416:
2411:
2410:
2395:
2380:
2378:
2377:
2372:
2367:
2366:
2351:
2334:is nonsingular.
2333:
2331:
2330:
2325:
2302:, implying that
2301:
2299:
2298:
2293:
2248:
2246:
2245:
2240:
2206:
2204:
2203:
2198:
2132:
2130:
2129:
2124:
2112:
2110:
2109:
2104:
2092:
2090:
2089:
2084:
2072:
2070:
2069:
2064:
2052:
2050:
2049:
2044:
2029:
2027:
2026:
2021:
2009:
2007:
2006:
2001:
1986:
1984:
1983:
1978:
1962:
1960:
1959:
1954:
1920:
1918:
1917:
1912:
1900:
1898:
1897:
1892:
1884:
1883:
1865:
1864:
1831:
1830:
1809:
1808:
1765:
1763:
1762:
1757:
1739:
1737:
1736:
1731:
1729:
1728:
1723:
1701:
1699:
1698:
1693:
1678:
1676:
1675:
1670:
1658:
1656:
1655:
1650:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1594:
1592:
1591:
1586:
1571:
1569:
1568:
1563:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1490:
1488:
1487:
1482:
1449:
1447:
1446:
1441:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1377:
1375:
1374:
1369:
1298:
1296:
1295:
1290:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1203:
1201:
1200:
1195:
1150:
1148:
1147:
1142:
1124:
1122:
1121:
1116:
1114:
1113:
1086:
1085:
1069:
1067:
1066:
1061:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
985:
983:
982:
977:
966:(i) Assume that
962:
960:
959:
954:
942:
940:
939:
934:
904:
902:
901:
896:
884:
882:
881:
876:
861:
859:
858:
853:
817:
815:
814:
809:
794:
792:
791:
786:
774:
772:
771:
766:
764:
763:
752:
733:
731:
730:
725:
723:
722:
711:
692:
690:
689:
684:
654:
652:
651:
646:
644:
643:
632:
613:
611:
610:
605:
603:
602:
591:
566:
564:
563:
558:
527:
525:
524:
519:
501:
499:
498:
493:
491:
490:
474:
472:
471:
466:
446:
444:
443:
438:
426:
424:
423:
418:
399:is of dimension
398:
396:
395:
390:
378:
376:
375:
370:
351:is of dimension
350:
348:
347:
342:
327:
325:
324:
319:
290:
289:
277:
276:
258:
257:
235:
233:
232:
227:
78:
76:
75:
70:
3635:
3634:
3630:
3629:
3628:
3626:
3625:
3624:
3605:
3604:
3590:Wayback Machine
3579:
3466:10.1.1.710.6894
3386:
3383:
3382:
3313:
3310:
3309:
3264:
3261:
3260:
3255:
3250:
3249:
3190:
3186:
3179:
3177:
3173:
3164:
3163:
3159:
3150:
3149:
3145:
3129:10.1.1.710.6894
3110:
3106:
3101:
3097:
3074:10.2307/3619888
3058:
3054:
3048:ill-conditioned
3045:
3041:
3036:
3032:
3015:
3011:
3006:
2988:
2979:
2978:. See also the
2971:
2946:
2942:
2933:
2932:
2930:
2927:
2926:
2897:
2894:
2893:
2877:
2874:
2873:
2866:
2837:
2836:
2831:
2825:
2824:
2819:
2809:
2808:
2798:
2797:
2791:
2787:
2785:
2779:
2778:
2770:
2764:
2760:
2753:
2752:
2745:
2744:
2739:
2733:
2732:
2727:
2717:
2716:
2709:
2708:
2702:
2698:
2696:
2690:
2689:
2684:
2678:
2674:
2667:
2666:
2664:
2661:
2660:
2600:
2599:
2594:
2588:
2587:
2582:
2572:
2571:
2569:
2566:
2565:
2547:
2546:
2541:
2535:
2534:
2529:
2519:
2518:
2516:
2513:
2512:
2511:to each other:
2487:, and a matrix
2469:
2430:
2427:
2426:
2406:
2402:
2388:
2386:
2383:
2382:
2362:
2358:
2344:
2342:
2339:
2338:
2307:
2304:
2303:
2254:
2251:
2250:
2216:
2213:
2212:
2138:
2135:
2134:
2118:
2115:
2114:
2098:
2095:
2094:
2078:
2075:
2074:
2058:
2055:
2054:
2035:
2032:
2031:
2015:
2012:
2011:
1992:
1989:
1988:
1972:
1969:
1968:
1936:
1933:
1932:
1906:
1903:
1902:
1879:
1875:
1860:
1856:
1826:
1822:
1804:
1800:
1771:
1768:
1767:
1745:
1742:
1741:
1724:
1719:
1718:
1707:
1704:
1703:
1684:
1681:
1680:
1664:
1661:
1660:
1644:
1641:
1640:
1620:
1617:
1616:
1600:
1597:
1596:
1577:
1574:
1573:
1557:
1554:
1553:
1528:
1525:
1524:
1496:
1493:
1492:
1455:
1452:
1451:
1432:
1429:
1428:
1412:
1409:
1408:
1383:
1380:
1379:
1312:
1309:
1308:
1269:
1266:
1265:
1249:
1246:
1245:
1229:
1226:
1225:
1209:
1206:
1205:
1156:
1153:
1152:
1130:
1127:
1126:
1109:
1105:
1081:
1077:
1075:
1072:
1071:
1034:
1031:
1030:
1014:
1011:
1010:
991:
988:
987:
971:
968:
967:
948:
945:
944:
910:
907:
906:
890:
887:
886:
867:
864:
863:
829:
826:
825:
800:
797:
796:
780:
777:
776:
775:if and only if
753:
748:
747:
739:
736:
735:
712:
707:
706:
698:
695:
694:
660:
657:
656:
633:
628:
627:
619:
616:
615:
592:
587:
586:
578:
575:
574:
573:Given matrices
540:
537:
536:
529:identity matrix
507:
504:
503:
486:
482:
480:
477:
476:
452:
449:
448:
432:
429:
428:
404:
401:
400:
384:
381:
380:
356:
353:
352:
336:
333:
332:
285:
281:
272:
268:
253:
249:
244:
241:
240:
221:
218:
217:
207:
104:complex numbers
43:
40:
39:
12:
11:
5:
3633:
3623:
3622:
3620:Control theory
3617:
3603:
3602:
3597:
3592:
3578:
3577:External links
3575:
3574:
3573:
3564:
3507:
3479:
3459:(2): 580â593.
3446:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3378:
3358:(9): 820â826.
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3303:
3280:
3277:
3274:
3271:
3268:
3254:
3251:
3248:
3247:
3184:
3171:
3157:
3143:
3122:(2): 580â593.
3104:
3095:
3052:
3039:
3030:
3008:
3007:
3005:
3002:
3001:
3000:
2995:
2987:
2984:
2954:
2949:
2945:
2941:
2936:
2901:
2881:
2865:
2862:
2858:
2857:
2846:
2841:
2835:
2832:
2830:
2827:
2826:
2823:
2820:
2818:
2815:
2814:
2812:
2807:
2802:
2794:
2790:
2786:
2784:
2781:
2780:
2777:
2774:
2771:
2767:
2763:
2759:
2758:
2756:
2749:
2743:
2740:
2738:
2735:
2734:
2731:
2728:
2726:
2723:
2722:
2720:
2713:
2705:
2701:
2697:
2695:
2692:
2691:
2688:
2685:
2681:
2677:
2673:
2672:
2670:
2604:
2598:
2595:
2593:
2590:
2589:
2586:
2583:
2581:
2578:
2577:
2575:
2551:
2545:
2542:
2540:
2537:
2536:
2533:
2530:
2528:
2525:
2524:
2522:
2468:
2465:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2414:
2409:
2405:
2401:
2398:
2394:
2391:
2370:
2365:
2361:
2357:
2354:
2350:
2347:
2323:
2320:
2317:
2314:
2311:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2122:
2102:
2082:
2062:
2042:
2039:
2019:
1999:
1996:
1976:
1952:
1949:
1946:
1943:
1940:
1910:
1890:
1887:
1882:
1878:
1874:
1871:
1868:
1863:
1859:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1829:
1825:
1821:
1818:
1815:
1812:
1807:
1803:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1755:
1752:
1749:
1727:
1722:
1717:
1714:
1711:
1691:
1688:
1668:
1648:
1624:
1604:
1584:
1581:
1561:
1538:
1535:
1532:
1512:
1509:
1506:
1503:
1500:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1439:
1436:
1416:
1396:
1393:
1390:
1387:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1288:
1285:
1282:
1279:
1276:
1273:
1253:
1233:
1213:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1140:
1137:
1134:
1112:
1108:
1104:
1101:
1098:
1095:
1092:
1089:
1084:
1080:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1018:
998:
995:
975:
952:
932:
929:
926:
923:
920:
917:
914:
894:
874:
871:
851:
848:
845:
842:
839:
836:
833:
807:
804:
784:
762:
759:
756:
751:
746:
743:
721:
718:
715:
710:
705:
702:
682:
679:
676:
673:
670:
667:
664:
642:
639:
636:
631:
626:
623:
601:
598:
595:
590:
585:
582:
556:
553:
550:
547:
544:
517:
514:
511:
489:
485:
464:
460:
456:
436:
416:
412:
408:
388:
368:
364:
360:
340:
329:
328:
317:
314:
311:
308:
305:
302:
299:
296:
293:
288:
284:
280:
275:
271:
267:
264:
261:
256:
252:
248:
225:
206:
203:
80:
79:
68:
65:
62:
59:
56:
53:
50:
47:
22:control theory
9:
6:
4:
3:
2:
3632:
3621:
3618:
3616:
3613:
3612:
3610:
3601:
3598:
3596:
3593:
3591:
3587:
3584:
3581:
3580:
3570:
3565:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3528:
3523:
3519:
3515:
3514:
3508:
3503:
3498:
3494:
3491:
3490:
3485:
3480:
3476:
3472:
3467:
3462:
3458:
3454:
3453:
3447:
3443:
3439:
3435:
3431:
3427:
3424:
3423:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3379:
3375:
3371:
3366:
3361:
3357:
3353:
3352:
3347:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3304:
3300:
3296:
3295:
3278:
3275:
3272:
3269:
3266:
3257:
3256:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3215:
3210:
3205:
3201:
3197:
3196:
3188:
3175:
3167:
3161:
3153:
3147:
3139:
3135:
3130:
3125:
3121:
3117:
3116:
3108:
3099:
3091:
3087:
3083:
3079:
3075:
3071:
3067:
3063:
3056:
3049:
3043:
3034:
3027:
3024: =
3023:
3020: â
3019:
3013:
3009:
2999:
2996:
2993:
2990:
2989:
2983:
2977:
2969:
2965:
2947:
2943:
2923:
2919:
2915:
2899:
2879:
2871:
2861:
2844:
2839:
2833:
2828:
2821:
2816:
2810:
2805:
2800:
2792:
2788:
2782:
2775:
2772:
2765:
2761:
2754:
2747:
2741:
2736:
2729:
2724:
2718:
2711:
2703:
2699:
2693:
2686:
2679:
2675:
2668:
2659:
2658:
2657:
2655:
2652: =
2651:
2648: â
2647:
2642:
2640:
2636:
2632:
2629: =
2628:
2625: â
2624:
2620:
2602:
2596:
2591:
2584:
2579:
2573:
2549:
2543:
2538:
2531:
2526:
2520:
2510:
2506:
2503: +
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2464:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2407:
2403:
2399:
2363:
2359:
2355:
2335:
2318:
2315:
2309:
2289:
2286:
2280:
2277:
2271:
2265:
2262:
2256:
2236:
2233:
2227:
2224:
2218:
2210:
2194:
2191:
2185:
2179:
2173:
2167:
2164:
2158:
2152:
2146:
2140:
2120:
2100:
2080:
2060:
2040:
2037:
2017:
1997:
1994:
1974:
1966:
1947:
1944:
1938:
1930:
1925:
1924:
1908:
1888:
1885:
1880:
1876:
1872:
1869:
1866:
1861:
1857:
1853:
1850:
1847:
1841:
1838:
1827:
1823:
1819:
1813:
1805:
1801:
1797:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1753:
1750:
1747:
1725:
1720:
1715:
1712:
1709:
1689:
1686:
1666:
1646:
1638:
1622:
1602:
1582:
1579:
1559:
1550:
1536:
1533:
1530:
1507:
1504:
1498:
1472:
1469:
1463:
1457:
1437:
1434:
1414:
1391:
1385:
1365:
1356:
1353:
1347:
1341:
1338:
1329:
1326:
1320:
1314:
1306:
1302:
1286:
1283:
1277:
1271:
1251:
1231:
1211:
1188:
1185:
1179:
1176:
1173:
1170:
1164:
1158:
1138:
1135:
1132:
1110:
1102:
1099:
1093:
1090:
1087:
1082:
1078:
1054:
1051:
1045:
1042:
1039:
1036:
1016:
996:
993:
973:
964:
950:
930:
927:
924:
921:
918:
915:
912:
892:
872:
869:
849:
846:
843:
840:
837:
834:
831:
824:The equation
823:
819:
805:
802:
782:
760:
757:
754:
744:
741:
719:
716:
713:
703:
700:
680:
677:
674:
671:
668:
665:
662:
640:
637:
634:
624:
621:
599:
596:
593:
583:
580:
572:
568:
554:
551:
548:
545:
542:
535:of dimension
534:
533:linear system
530:
515:
512:
509:
487:
483:
462:
458:
454:
447:of dimension
434:
414:
410:
406:
386:
366:
362:
358:
338:
315:
312:
309:
306:
303:
300:
297:
294:
286:
282:
278:
273:
269:
265:
262:
259:
254:
250:
239:
238:
237:
223:
216:
212:
202:
200:
196:
192:
188:
184:
180:
176:
172:
168:
165: =
164:
161: +
160:
156:
152:
148:
144:
139:
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
66:
63:
60:
57:
54:
51:
48:
45:
38:
37:
36:
35:of the form:
34:
31:
27:
23:
19:
3568:
3517:
3511:
3492:
3487:
3456:
3450:
3425:
3420:
3355:
3349:
3298:
3292:
3199:
3193:
3187:
3174:
3160:
3146:
3119:
3113:
3107:
3098:
3065:
3061:
3055:
3042:
3033:
3025:
3021:
3017:
3012:
2974:function in
2918:QR algorithm
2867:
2859:
2653:
2649:
2645:
2643:
2638:
2634:
2630:
2626:
2622:
2618:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2472:
2470:
2336:
1926:
1922:
1551:
965:
821:
820:
570:
569:
330:
208:
194:
190:
182:
178:
175:Banach space
166:
162:
158:
154:
150:
142:
140:
135:
131:
127:
123:
119:
115:
111:
107:
99:
95:
91:
87:
81:
25:
15:
3428:(1): 1â21.
1637:eigenvector
1299:due to the
147:eigenvalues
18:mathematics
3609:Categories
3527:1502.03121
3419: ?".
3253:References
3209:1502.03121
2976:GNU Octave
2914:Schur form
2621:such that
2479:, of size
2133:such that
209:Using the
153:and â
130:both have
3461:CiteSeerX
3442:122259404
3395:−
3351:Comm. ACM
3124:CiteSeerX
3090:126229881
2980:sylvester
2773:−
2408:∗
2364:∗
2316:−
2278:−
2263:−
2225:−
2207:. By the
2192:≡
2038:−
1995:−
1945:−
1881:∗
1870:λ
1867:−
1862:∗
1851:λ
1839:−
1828:∗
1814:−
1806:∗
1751:≠
1726:∗
1687:−
1603:λ
1580:−
1505:−
1470:−
1464:σ
1435:−
1392:⋅
1386:σ
1354:−
1348:σ
1327:−
1315:σ
1307:tells us
1186:−
1136:≥
1125:for each
1100:−
1052:−
994:−
803:−
758:×
745:∈
717:×
704:∈
638:×
625:∈
597:×
584:∈
549:×
513:×
459:×
411:×
363:×
310:
298:
279:⊗
260:⊗
138:columns.
134:rows and
3615:Matrices
3586:Archived
3552:26208345
3374:12957010
3234:26208345
2986:See also
2970:and the
2491:of size
2010:. Since
1264:. Then
734:for any
571:Theorem.
199:disjoint
33:equation
3532:Bibcode
3214:Bibcode
3082:3619888
2509:similar
2249:. Thus
1766:, and
1740:. Then
502:is the
187:spectra
3560:665111
3558:
3550:
3463:
3440:
3372:
3242:665111
3240:
3232:
3126:
3088:
3080:
2968:LAPACK
2656:then
1923:Q.E.D.
1901:Hence
1702:, and
1615:. Let
1378:where
822:Proof.
331:where
94:, and
30:matrix
3556:S2CID
3522:arXiv
3438:S2CID
3370:S2CID
3238:S2CID
3204:arXiv
3086:S2CID
3078:JSTOR
3004:Notes
2916:by a
2912:into
193:and â
28:is a
3548:PMID
3230:PMID
3195:IEEE
3178:The
2972:lyap
2892:and
2564:and
2507:are
2483:and
2475:and
2381:and
2113:and
2073:and
2030:and
1639:for
1572:and
1427:and
986:and
795:and
614:and
475:and
197:are
126:and
118:and
110:and
24:, a
3540:doi
3497:doi
3493:435
3471:doi
3430:doi
3360:doi
3291:".
3222:doi
3180:syl
3134:doi
3070:doi
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307:vec
295:vec
224:vec
189:of
149:of
16:In
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3546:.
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3516:.
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3299:99
3297:.
3236:.
3228:.
3220:.
3212:.
3200:24
3198:.
3132:.
3120:36
3118:.
3084:.
3076:.
3066:82
3064:.
3022:XB
3018:AX
2650:XB
2646:AX
2641:.
2627:XB
2623:AX
2211:,
1889:0.
1659:,
963:.
567:.
427:,
379:,
201:.
163:XB
159:AX
90:,
3562:.
3542::
3534::
3524::
3505:.
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3477:.
3473::
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3407:Y
3404:=
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3398:X
3392:X
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3376:.
3362::
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3331:=
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3325:X
3322:+
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3276:x
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3216::
3206::
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3140:.
3136::
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3028:.
3026:C
2953:)
2948:3
2944:n
2940:(
2935:O
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2840:]
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2817:A
2811:[
2806:=
2801:]
2793:m
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2755:[
2748:]
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2712:]
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2603:]
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2574:[
2550:]
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2521:[
2505:m
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2497:m
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1942:(
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1836:(
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1817:(
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1467:(
1461:(
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1111:k
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1103:B
1097:(
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925:B
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847:=
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841:X
838:+
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783:A
761:m
755:n
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709:C
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681:C
678:=
675:B
672:X
669:+
666:X
663:A
641:m
635:m
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600:n
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247:(
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179:X
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112:B
108:A
100:X
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92:B
88:A
67:.
64:C
61:=
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55:X
52:+
49:X
46:A
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