1831:
609:
1867:
279:
604:{\displaystyle P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.}
210:
746:
that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational
715:
664:
271:
242:
60:
1489:
1703:
922:
1794:
1908:
1713:
1479:
1514:
1061:
1278:
915:
1353:
1509:
1031:
769:
1901:
1613:
1484:
1398:
824:
1718:
1608:
1316:
996:
848:
Friedland, S.; Melman, A. (2020). "A note on
Hermitian positive semidefinite matrix polynomials".
1753:
1682:
1564:
1424:
1021:
908:
669:
1623:
1206:
1011:
624:
1937:
1569:
1306:
1156:
1151:
986:
961:
956:
759:
1932:
1894:
1882:
1874:
1763:
1121:
951:
931:
36:
247:
218:
8:
1784:
1758:
1336:
1141:
1131:
788:
731:
1835:
1789:
1779:
1733:
1728:
1657:
1593:
1459:
1196:
1191:
1126:
1116:
981:
615:
1927:
1846:
1830:
1633:
1628:
1618:
1598:
1559:
1554:
1383:
1378:
1363:
1358:
1349:
1344:
1291:
1186:
1081:
1051:
1046:
1026:
1016:
976:
888:
772:(semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.
739:
718:
43:. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.
17:
784:, which are simply matrices with exactly one non-zero entry in each row and column.
1841:
1809:
1738:
1677:
1672:
1652:
1588:
1494:
1464:
1449:
1429:
1368:
1321:
1296:
1286:
1257:
1176:
1171:
1146:
1076:
1056:
966:
946:
880:
857:
766:
1434:
1539:
1474:
1454:
1439:
1419:
1403:
1301:
1232:
1222:
1181:
1066:
1036:
781:
751:
205:{\displaystyle P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p}}
1878:
1799:
1743:
1723:
1708:
1667:
1544:
1504:
1469:
1393:
1332:
1311:
1252:
1242:
1227:
1161:
1106:
1096:
1091:
1001:
743:
884:
861:
1921:
1804:
1662:
1603:
1534:
1524:
1519:
1444:
1373:
1247:
1237:
1166:
1086:
1071:
1006:
892:
755:
1687:
1644:
1549:
1262:
1201:
1111:
991:
1529:
1499:
1267:
1101:
971:
735:
24:
1580:
1041:
40:
1814:
1388:
1748:
900:
1866:
273:
is non-zero. An example 3Ă—3 polynomial matrix, degree 2:
530:
461:
392:
294:
672:
627:
282:
250:
221:
63:
738:
equal to a non-zero element of that field is called
709:
658:
603:
265:
236:
204:
1919:
847:
244:denotes a matrix of constant coefficients, and
39:whose elements are univariate or multivariate
1902:
916:
874:
765:The determinant of a matrix polynomial with
1909:
1895:
1490:Fundamental (linear differential equation)
923:
909:
750:The roots of a polynomial matrix over the
877:Error-free Polynomial Matrix computations
799:be a polynomial matrix, then the matrix λ
791:over which we constructed the matrix, by
614:We can express this by saying that for a
1795:Matrix representation of conic sections
1920:
904:
787:If by λ we denote any element of the
1861:
850:Linear Algebra and Its Applications
13:
930:
776:Note that polynomial matrices are
14:
1949:
1865:
1829:
795:the identity matrix, and we let
1697:Used in science and engineering
46:A univariate polynomial matrix
940:Explicitly constrained entries
841:
704:
698:
695:
692:
686:
673:
653:
650:
644:
638:
260:
254:
231:
225:
189:
183:
158:
152:
140:
134:
125:
119:
100:
94:
1:
1714:Fundamental (computer vision)
834:
724:
1881:. You can help Knowledge by
875:Krishnamurthy, E.V. (1985).
7:
1480:Duplication and elimination
1279:eigenvalues or eigenvectors
730:A polynomial matrix over a
10:
1954:
1860:
1413:With specific applications
1042:Discrete Fourier Transform
710:{\displaystyle (M_{n}(R))}
15:
1823:
1772:
1704:Cabibbo–Kobayashi–Maskawa
1696:
1642:
1578:
1412:
1331:Satisfying conditions on
1330:
1276:
1215:
939:
885:10.1007/978-1-4612-5118-7
862:10.1016/j.laa.2020.03.038
825:characteristic polynomial
659:{\displaystyle M_{n}(R)}
16:Not to be confused with
1062:Generalized permutation
758:where the matrix loses
1836:Mathematics portal
754:are the points in the
711:
660:
605:
267:
238:
206:
90:
815:. Its determinant, |λ
809:characteristic matrix
712:
661:
606:
268:
239:
207:
70:
33:matrix of polynomials
780:to be confused with
670:
625:
280:
266:{\displaystyle A(p)}
248:
237:{\displaystyle A(i)}
219:
61:
1873:This article about
1785:Linear independence
1032:Diagonally dominant
827:of the matrix
819: −
803: −
1790:Matrix exponential
1780:Jordan normal form
1614:Fisher information
1485:Euclidean distance
1399:Totally unimodular
707:
656:
601:
582:
513:
447:
378:
263:
234:
202:
1890:
1889:
1855:
1854:
1847:Category:Matrices
1719:Fuzzy associative
1609:Doubly stochastic
1317:Positive-definite
997:Block tridiagonal
782:monomial matrices
770:positive-definite
29:polynomial matrix
18:matrix polynomial
1945:
1911:
1904:
1897:
1869:
1862:
1842:List of matrices
1834:
1833:
1810:Row echelon form
1754:State transition
1683:Seidel adjacency
1565:Totally positive
1425:Alternating sign
1022:Complex Hadamard
925:
918:
911:
902:
901:
896:
866:
865:
845:
716:
714:
713:
708:
685:
684:
665:
663:
662:
657:
637:
636:
610:
608:
607:
602:
597:
596:
587:
586:
518:
517:
452:
451:
383:
382:
364:
363:
311:
310:
272:
270:
269:
264:
243:
241:
240:
235:
211:
209:
208:
203:
201:
200:
170:
169:
112:
111:
89:
84:
1953:
1952:
1948:
1947:
1946:
1944:
1943:
1942:
1918:
1917:
1916:
1915:
1858:
1856:
1851:
1828:
1819:
1768:
1692:
1638:
1574:
1408:
1326:
1272:
1211:
1012:Centrosymmetric
935:
929:
899:
870:
869:
846:
842:
837:
752:complex numbers
727:
680:
676:
671:
668:
667:
632:
628:
626:
623:
622:
592:
588:
581:
580:
575:
570:
564:
563:
558:
553:
547:
546:
541:
536:
526:
525:
512:
511:
506:
501:
495:
494:
489:
484:
478:
477:
472:
467:
457:
456:
446:
445:
440:
432:
426:
425:
420:
415:
409:
408:
403:
398:
388:
387:
377:
376:
371:
359:
355:
353:
338:
337:
332:
324:
318:
317:
312:
306:
302:
300:
290:
289:
281:
278:
277:
249:
246:
245:
220:
217:
216:
196:
192:
165:
161:
107:
103:
85:
74:
62:
59:
58:
54:is defined as:
21:
12:
11:
5:
1951:
1941:
1940:
1935:
1930:
1914:
1913:
1906:
1899:
1891:
1888:
1887:
1870:
1853:
1852:
1850:
1849:
1844:
1839:
1824:
1821:
1820:
1818:
1817:
1812:
1807:
1802:
1800:Perfect matrix
1797:
1792:
1787:
1782:
1776:
1774:
1770:
1769:
1767:
1766:
1761:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1700:
1698:
1694:
1693:
1691:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1649:
1647:
1640:
1639:
1637:
1636:
1631:
1626:
1621:
1616:
1611:
1606:
1601:
1596:
1591:
1585:
1583:
1576:
1575:
1573:
1572:
1570:Transformation
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1452:
1447:
1442:
1437:
1432:
1427:
1422:
1416:
1414:
1410:
1409:
1407:
1406:
1401:
1396:
1391:
1386:
1381:
1376:
1371:
1366:
1361:
1356:
1347:
1341:
1339:
1328:
1327:
1325:
1324:
1319:
1314:
1309:
1307:Diagonalizable
1304:
1299:
1294:
1289:
1283:
1281:
1277:Conditions on
1274:
1273:
1271:
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1225:
1219:
1217:
1213:
1212:
1210:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1159:
1157:Skew-symmetric
1154:
1152:Skew-Hermitian
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
987:Block-diagonal
984:
979:
974:
969:
964:
962:Anti-symmetric
959:
957:Anti-Hermitian
954:
949:
943:
941:
937:
936:
928:
927:
920:
913:
905:
898:
897:
871:
868:
867:
839:
838:
836:
833:
811:of the matrix
774:
773:
763:
748:
726:
723:
706:
703:
700:
697:
694:
691:
688:
683:
679:
675:
655:
652:
649:
646:
643:
640:
635:
631:
612:
611:
600:
595:
591:
585:
579:
576:
574:
571:
569:
566:
565:
562:
559:
557:
554:
552:
549:
548:
545:
542:
540:
537:
535:
532:
531:
529:
524:
521:
516:
510:
507:
505:
502:
500:
497:
496:
493:
490:
488:
485:
483:
480:
479:
476:
473:
471:
468:
466:
463:
462:
460:
455:
450:
444:
441:
439:
436:
433:
431:
428:
427:
424:
421:
419:
416:
414:
411:
410:
407:
404:
402:
399:
397:
394:
393:
391:
386:
381:
375:
372:
370:
367:
362:
358:
354:
352:
349:
346:
343:
340:
339:
336:
333:
331:
328:
325:
323:
320:
319:
316:
313:
309:
305:
301:
299:
296:
295:
293:
288:
285:
262:
259:
256:
253:
233:
230:
227:
224:
213:
212:
199:
195:
191:
188:
185:
182:
179:
176:
173:
168:
164:
160:
157:
154:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
121:
118:
115:
110:
106:
102:
99:
96:
93:
88:
83:
80:
77:
73:
69:
66:
9:
6:
4:
3:
2:
1950:
1939:
1936:
1934:
1931:
1929:
1926:
1925:
1923:
1912:
1907:
1905:
1900:
1898:
1893:
1892:
1886:
1884:
1880:
1876:
1871:
1868:
1864:
1863:
1859:
1848:
1845:
1843:
1840:
1838:
1837:
1832:
1826:
1825:
1822:
1816:
1813:
1811:
1808:
1806:
1805:Pseudoinverse
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1777:
1775:
1773:Related terms
1771:
1765:
1764:Z (chemistry)
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1701:
1699:
1695:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1650:
1648:
1646:
1641:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1586:
1584:
1582:
1577:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1417:
1415:
1411:
1405:
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1351:
1348:
1346:
1343:
1342:
1340:
1338:
1334:
1329:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1284:
1282:
1280:
1275:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1220:
1218:
1214:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1122:Pentadiagonal
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
952:Anti-diagonal
950:
948:
945:
944:
942:
938:
933:
926:
921:
919:
914:
912:
907:
906:
903:
894:
890:
886:
882:
878:
873:
872:
863:
859:
855:
851:
844:
840:
832:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
785:
783:
779:
771:
768:
764:
761:
757:
756:complex plane
753:
749:
745:
742:, and has an
741:
737:
733:
729:
728:
722:
720:
701:
689:
681:
677:
647:
641:
633:
629:
620:
617:
598:
593:
589:
583:
577:
572:
567:
560:
555:
550:
543:
538:
533:
527:
522:
519:
514:
508:
503:
498:
491:
486:
481:
474:
469:
464:
458:
453:
448:
442:
437:
434:
429:
422:
417:
412:
405:
400:
395:
389:
384:
379:
373:
368:
365:
360:
356:
350:
347:
344:
341:
334:
329:
326:
321:
314:
307:
303:
297:
291:
286:
283:
276:
275:
274:
257:
251:
228:
222:
197:
193:
186:
180:
177:
174:
171:
166:
162:
155:
149:
146:
143:
137:
131:
128:
122:
116:
113:
108:
104:
97:
91:
86:
81:
78:
75:
71:
67:
64:
57:
56:
55:
53:
49:
44:
42:
38:
34:
30:
26:
19:
1938:Matrix stubs
1883:expanding it
1872:
1857:
1827:
1759:Substitution
1645:graph theory
1142:Quaternionic
1136:
1132:Persymmetric
879:. Springer.
876:
853:
849:
843:
828:
820:
816:
812:
808:
804:
800:
796:
792:
786:
777:
775:
621:, the rings
618:
613:
214:
51:
47:
45:
32:
28:
22:
1933:Polynomials
1734:Hamiltonian
1658:Biadjacency
1594:Correlation
1510:Householder
1460:Commutation
1197:Vandermonde
1192:Tridiagonal
1127:Permutation
1117:Nonnegative
1102:Matrix unit
982:Bisymmetric
856:: 105–109.
736:determinant
41:polynomials
25:mathematics
1922:Categories
1634:Transition
1629:Stochastic
1599:Covariance
1581:statistics
1560:Symplectic
1555:Similarity
1384:Unimodular
1379:Orthogonal
1364:Involutory
1359:Invertible
1354:Projection
1350:Idempotent
1292:Convergent
1187:Triangular
1137:Polynomial
1082:Hessenberg
1052:Equivalent
1047:Elementary
1027:Copositive
1017:Conference
977:Bidiagonal
835:References
740:unimodular
725:Properties
719:isomorphic
50:of degree
1815:Wronskian
1739:Irregular
1729:Gell-Mann
1678:Laplacian
1673:Incidence
1653:Adjacency
1624:Precision
1589:Centering
1495:Generator
1465:Confusion
1450:Circulant
1430:Augmented
1389:Unipotent
1369:Nilpotent
1345:Congruent
1322:Stieltjes
1297:Defective
1287:Companion
1258:Redheffer
1177:Symmetric
1172:Sylvester
1147:Signature
1077:Hermitian
1057:Frobenius
967:Arrowhead
947:Alternant
893:858879932
823:| is the
767:Hermitian
747:function.
435:−
366:−
175:⋯
72:∑
1928:Matrices
1875:matrices
1643:Used in
1579:Used in
1540:Rotation
1515:Jacobian
1475:Distance
1455:Cofactor
1440:Carleman
1420:Adjugate
1404:Weighing
1337:inverses
1333:products
1302:Definite
1233:Identity
1223:Exchange
1216:Constant
1182:Toeplitz
1067:Hadamard
1037:Diagonal
1744:Overlap
1709:Density
1668:Edmonds
1545:Seifert
1505:Hessian
1470:Coxeter
1394:Unitary
1312:Hurwitz
1243:Of ones
1228:Hilbert
1162:Skyline
1107:Metzler
1097:Logical
1092:Integer
1002:Boolean
934:classes
807:is the
744:inverse
1663:Degree
1604:Design
1535:Random
1525:Payoff
1520:Moment
1445:Cartan
1435:BĂ©zout
1374:Normal
1248:Pascal
1238:Lehmer
1167:Sparse
1087:Hollow
1072:Hankel
1007:Cauchy
932:Matrix
891:
215:where
37:matrix
1877:is a
1724:Gamma
1688:Tutte
1550:Shear
1263:Shift
1253:Pauli
1202:Walsh
1112:Moore
992:Block
789:field
734:with
732:field
35:is a
1879:stub
1530:Pick
1500:Gram
1268:Zero
972:Band
889:OCLC
760:rank
717:are
666:and
616:ring
27:, a
1619:Hat
1352:or
1335:or
881:doi
858:doi
854:598
778:not
31:or
23:In
1924::
887:.
852:.
831:.
721:.
1910:e
1903:t
1896:v
1885:.
1749:S
1207:Z
924:e
917:t
910:v
895:.
883::
864:.
860::
829:A
821:A
817:I
813:A
805:A
801:I
797:A
793:I
762:.
705:]
702:X
699:[
696:)
693:)
690:R
687:(
682:n
678:M
674:(
654:)
651:]
648:X
645:[
642:R
639:(
634:n
630:M
619:R
599:.
594:2
590:x
584:)
578:0
573:1
568:0
561:0
556:0
551:0
544:0
539:1
534:0
528:(
523:+
520:x
515:)
509:0
504:0
499:3
492:0
487:2
482:0
475:1
470:0
465:0
459:(
454:+
449:)
443:0
438:1
430:2
423:2
418:0
413:0
406:0
401:0
396:1
390:(
385:=
380:)
374:0
369:1
361:2
357:x
351:2
348:+
345:x
342:3
335:2
330:x
327:2
322:0
315:x
308:2
304:x
298:1
292:(
287:=
284:P
261:)
258:p
255:(
252:A
232:)
229:i
226:(
223:A
198:p
194:x
190:)
187:p
184:(
181:A
178:+
172:+
167:2
163:x
159:)
156:2
153:(
150:A
147:+
144:x
141:)
138:1
135:(
132:A
129:+
126:)
123:0
120:(
117:A
114:=
109:n
105:x
101:)
98:n
95:(
92:A
87:p
82:0
79:=
76:n
68:=
65:P
52:p
48:P
20:.
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