2232:
20:
1042:, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. A similar result holds for
931:
201:
505:
338:
420:
778:
811:
84:
427:
229:
1890:
369:
1133:"A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices"
2104:
1323:
2195:
1195:
Yasuda, Mark (2003). "A Spectral
Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices".
1256:
Weaver, James R. (1985). "Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors".
2114:
1880:
728:
1246:
616:
1915:
1462:
1679:
1316:
27:
This article is about a matrix symmetric about its center. For a matrix symmetric about its diagonal, see
1754:
1226:
1910:
1432:
2014:
1885:
1799:
258:
2119:
2009:
1717:
1397:
2268:
2154:
2083:
1965:
1825:
1422:
1309:
2024:
1607:
1238:
926:{\displaystyle A_{i,\,j}=-A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.}
1970:
1707:
1557:
1552:
1387:
1362:
1357:
196:{\displaystyle A_{i,\,j}=A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.}
2164:
1522:
1352:
1332:
51:
43:
1291:
500:{\displaystyle {\begin{bmatrix}a&b&c\\d&e&d\\c&b&a\end{bmatrix}}.}
8:
2185:
2159:
1737:
1542:
1532:
620:
544:
1164:"Characterization and properties of matrices with generalized symmetry or skew symmetry"
333:{\displaystyle J_{i,\,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}}
2236:
2190:
2180:
2134:
2129:
2058:
1994:
1860:
1597:
1592:
1527:
1517:
1382:
1273:
1231:
1031:
699:
1081:
1068:
Yasuda, Mark (2012). "Some properties of commuting and anti-commuting m-involutions".
2273:
2247:
2231:
2034:
2029:
2019:
1999:
1960:
1955:
1784:
1779:
1764:
1759:
1750:
1745:
1692:
1587:
1537:
1482:
1452:
1447:
1427:
1417:
1377:
1242:
1113:
1096:
969:
598:
2242:
2210:
2139:
2078:
2073:
2053:
1989:
1895:
1865:
1850:
1830:
1769:
1722:
1697:
1687:
1658:
1577:
1572:
1547:
1477:
1457:
1367:
1347:
1265:
1204:
1175:
1144:
1108:
1077:
1043:
1027:
509:
28:
1835:
1940:
1875:
1855:
1840:
1820:
1804:
1702:
1633:
1623:
1582:
1467:
1437:
594:
512:
219:
2200:
2144:
2124:
2109:
2068:
1945:
1905:
1870:
1794:
1733:
1712:
1653:
1643:
1628:
1562:
1507:
1497:
1492:
1402:
573:
345:
39:
1208:
1180:
1163:
1148:
2262:
2205:
2063:
2004:
1935:
1925:
1920:
1845:
1774:
1648:
1638:
1567:
1487:
1472:
1407:
2088:
2045:
1950:
1663:
1602:
1512:
1392:
1035:
223:
1930:
1900:
1668:
1502:
1372:
1039:
652:
35:
1981:
1442:
1277:
695:
645:
19:
2215:
1789:
1295:
1269:
1132:
2149:
1301:
1001:
415:{\displaystyle {\begin{bmatrix}a&b\\b&a\end{bmatrix}}.}
23:
Symmetry pattern of a centrosymmetric 5 × 5 matrix
752:
424:
All 3 × 3 centrosymmetric matrices have the form
366:
All 2 × 2 centrosymmetric matrices have the form
326:
436:
378:
814:
731:
519:
430:
372:
232:
87:
1010:. The inverse problem for the commutation relation
1046:centrosymmetric and skew-centrosymmetric matrices.
1230:
925:
772:
499:
414:
332:
195:
2260:
964:lends itself to a natural generalization, where
773:{\displaystyle {\frac {m^{2}+m{\bmod {2}}}{2}}.}
655:can each be chosen so that they satisfy either
1030:centrosymmetric matrices are sometimes called
1317:
597:is also centrosymmetric, it follows that the
917:
897:
709:The maximum number of unique elements in an
187:
167:
79:is centrosymmetric when its entries satisfy
951:is the exchange matrix defined previously.
1891:Fundamental (linear differential equation)
1324:
1310:
694:is a centrosymmetric matrix with distinct
1179:
1130:
1112:
913:
906:
861:
826:
244:
183:
176:
131:
99:
1233:A Treatise on the Theory of Determinants
1126:
1124:
18:
2196:Matrix representation of conic sections
2261:
1255:
1194:
1161:
1094:
1067:
1063:
1061:
1059:
54:which is symmetric about its center.
1305:
1188:
1121:
785:
1225:
640:is a centrosymmetric matrix with an
57:
1056:
13:
1331:
1219:
1155:
1097:"Eigenvectors of certain matrices"
520:Algebraic structure and properties
14:
2285:
1285:
1131:Tao, David; Yasuda, Mark (2002).
1088:
1020:that commute with a fixed matrix
16:Matrix symmetric about its center
2230:
543:centrosymmetric matrices over a
2098:Used in science and engineering
986:) or, more generally, a matrix
879:
149:
1341:Explicitly constrained entries
1016:of identifying all involutory
611:centrosymmetric matrices over
1:
2115:Fundamental (computer vision)
1258:American Mathematical Monthly
1082:10.1016/S0252-9602(12)60044-7
1049:
954:The centrosymmetric relation
515:matrices are centrosymmetric.
1114:10.1016/0024-3795(73)90049-9
7:
1881:Duplication and elimination
1680:eigenvalues or eigenvectors
937:is skew-centrosymmetric if
360:
10:
2290:
1814:With specific applications
1443:Discrete Fourier Transform
579:is centrosymmetric, since
26:
2224:
2173:
2105:Cabibbo–Kobayashi–Maskawa
2097:
2043:
1979:
1813:
1732:Satisfying conditions on
1731:
1677:
1616:
1340:
1209:10.1137/S0895479802418835
1197:SIAM J. Matrix Anal. Appl
1181:10.1016/j.laa.2003.07.013
1149:10.1137/S0895479801386730
1137:SIAM J. Matrix Anal. Appl
1070:Acta Mathematica Scientia
719:centrosymmetric matrix is
698:, then the matrices that
808:if its entries satisfy
706:must be centrosymmetric.
1463:Generalized permutation
1024:has also been studied.
687:is the exchange matrix.
2237:Mathematics portal
1292:Centrosymmetric matrix
1162:Trench, W. F. (2004).
927:
774:
501:
416:
334:
197:
48:centrosymmetric matrix
24:
1095:Andrew, Alan (1973).
928:
775:
502:
417:
335:
198:
22:
1032:bisymmetric matrices
968:is replaced with an
812:
806:skew-centrosymmetric
729:
428:
370:
230:
85:
2186:Linear independence
1433:Diagonally dominant
1168:Linear Algebra Appl
1101:Linear Algebra Appl
621:associative algebra
344:is centrosymmetric
2191:Matrix exponential
2181:Jordan normal form
2015:Fisher information
1886:Euclidean distance
1800:Totally unimodular
923:
786:Related structures
770:
497:
488:
412:
403:
330:
325:
226:and 0 elsewhere:
205:Alternatively, if
193:
25:
2256:
2255:
2248:Category:Matrices
2120:Fuzzy associative
2010:Doubly stochastic
1718:Positive-definite
1398:Block tridiagonal
1237:. Dover. p.
970:involutory matrix
883:
765:
153:
58:Formal definition
2281:
2243:List of matrices
2235:
2234:
2211:Row echelon form
2155:State transition
2084:Seidel adjacency
1966:Totally positive
1826:Alternating sign
1423:Complex Hadamard
1326:
1319:
1312:
1303:
1302:
1281:
1252:
1236:
1213:
1212:
1192:
1186:
1185:
1183:
1159:
1153:
1152:
1128:
1119:
1118:
1116:
1092:
1086:
1085:
1065:
1023:
1019:
1015:
1009:
999:
989:
985:
974:
967:
963:
950:
946:
936:
932:
930:
929:
924:
884:
881:
878:
877:
831:
830:
803:
799:
779:
777:
776:
771:
766:
761:
760:
759:
744:
743:
733:
718:
705:
693:
686:
682:
668:
651:
643:
639:
632:
614:
610:
592:
578:
572:. Moreover, the
571:
567:
563:
559:
549:
542:
532:
528:
506:
504:
503:
498:
493:
492:
421:
419:
418:
413:
408:
407:
356:
343:
339:
337:
336:
331:
329:
328:
249:
248:
218:
208:
202:
200:
199:
194:
154:
151:
148:
147:
104:
103:
78:
71:
38:, especially in
29:Symmetric matrix
2289:
2288:
2284:
2283:
2282:
2280:
2279:
2278:
2259:
2258:
2257:
2252:
2229:
2220:
2169:
2093:
2039:
1975:
1809:
1727:
1673:
1612:
1413:Centrosymmetric
1336:
1330:
1288:
1270:10.2307/2323222
1264:(10): 711–717.
1249:
1222:
1220:Further reading
1217:
1216:
1193:
1189:
1160:
1156:
1129:
1122:
1093:
1089:
1066:
1057:
1052:
1021:
1017:
1011:
1004:
991:
987:
976:
972:
965:
955:
948:
938:
934:
880:
842:
838:
819:
815:
813:
810:
809:
801:
795: ×
791:
788:
755:
751:
739:
735:
734:
732:
730:
727:
726:
714: ×
710:
703:
691:
684:
670:
656:
649:
641:
637:
628: ×
624:
612:
606: ×
602:
595:identity matrix
580:
576:
569:
565:
561:
551:
547:
538: ×
534:
530:
526:
522:
487:
486:
481:
476:
470:
469:
464:
459:
453:
452:
447:
442:
432:
431:
429:
426:
425:
402:
401:
396:
390:
389:
384:
374:
373:
371:
368:
367:
363:
348:
341:
324:
323:
300:
291:
290:
267:
254:
253:
237:
233:
231:
228:
227:
220:exchange matrix
210:
206:
150:
112:
108:
92:
88:
86:
83:
82:
73:
67: ×
63:
60:
32:
17:
12:
11:
5:
2287:
2277:
2276:
2271:
2269:Linear algebra
2254:
2253:
2251:
2250:
2245:
2240:
2225:
2222:
2221:
2219:
2218:
2213:
2208:
2203:
2201:Perfect matrix
2198:
2193:
2188:
2183:
2177:
2175:
2171:
2170:
2168:
2167:
2162:
2157:
2152:
2147:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2101:
2099:
2095:
2094:
2092:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2050:
2048:
2041:
2040:
2038:
2037:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1986:
1984:
1977:
1976:
1974:
1973:
1971:Transformation
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1817:
1815:
1811:
1810:
1808:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1748:
1742:
1740:
1729:
1728:
1726:
1725:
1720:
1715:
1710:
1708:Diagonalizable
1705:
1700:
1695:
1690:
1684:
1682:
1678:Conditions on
1675:
1674:
1672:
1671:
1666:
1661:
1656:
1651:
1646:
1641:
1636:
1631:
1626:
1620:
1618:
1614:
1613:
1611:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1558:Skew-symmetric
1555:
1553:Skew-Hermitian
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1395:
1390:
1388:Block-diagonal
1385:
1380:
1375:
1370:
1365:
1363:Anti-symmetric
1360:
1358:Anti-Hermitian
1355:
1350:
1344:
1342:
1338:
1337:
1329:
1328:
1321:
1314:
1306:
1300:
1299:
1287:
1286:External links
1284:
1283:
1282:
1253:
1247:
1221:
1218:
1215:
1214:
1203:(3): 601–605.
1187:
1154:
1143:(3): 885–895.
1120:
1107:(2): 151–162.
1087:
1076:(2): 631–644.
1054:
1053:
1051:
1048:
933:Equivalently,
922:
919:
916:
912:
909:
905:
902:
899:
896:
893:
890:
887:
876:
873:
870:
867:
864:
860:
857:
854:
851:
848:
845:
841:
837:
834:
829:
825:
822:
818:
804:is said to be
787:
784:
783:
782:
781:
780:
769:
764:
758:
754:
750:
747:
742:
738:
721:
720:
707:
688:
634:
574:matrix product
550:, then so are
521:
518:
517:
516:
507:
496:
491:
485:
482:
480:
477:
475:
472:
471:
468:
465:
463:
460:
458:
455:
454:
451:
448:
446:
443:
441:
438:
437:
435:
422:
411:
406:
400:
397:
395:
392:
391:
388:
385:
383:
380:
379:
377:
362:
359:
346:if and only if
340:then a matrix
327:
322:
319:
316:
313:
310:
307:
304:
301:
299:
296:
293:
292:
289:
286:
283:
280:
277:
274:
271:
268:
266:
263:
260:
259:
257:
252:
247:
243:
240:
236:
222:with 1 on the
192:
189:
186:
182:
179:
175:
172:
169:
166:
163:
160:
157:
146:
143:
140:
137:
134:
130:
127:
124:
121:
118:
115:
111:
107:
102:
98:
95:
91:
59:
56:
40:linear algebra
15:
9:
6:
4:
3:
2:
2286:
2275:
2272:
2270:
2267:
2266:
2264:
2249:
2246:
2244:
2241:
2239:
2238:
2233:
2227:
2226:
2223:
2217:
2214:
2212:
2209:
2207:
2206:Pseudoinverse
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2178:
2176:
2174:Related terms
2172:
2166:
2165:Z (chemistry)
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2102:
2100:
2096:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2051:
2049:
2047:
2042:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1987:
1985:
1983:
1978:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1818:
1816:
1812:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1752:
1749:
1747:
1744:
1743:
1741:
1739:
1735:
1730:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1685:
1683:
1681:
1676:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1621:
1619:
1615:
1609:
1606:
1604:
1601:
1599:
1596:
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:
1523:Pentadiagonal
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:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
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:
1353:Anti-diagonal
1351:
1349:
1346:
1345:
1343:
1339:
1334:
1327:
1322:
1320:
1315:
1313:
1308:
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958:
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920:
914:
910:
907:
903:
900:
894:
891:
888:
885:
882:for all
874:
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832:
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767:
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722:
717:
713:
708:
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697:
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673:
667:
663:
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644:-dimensional
635:
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627:
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583:
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554:
546:
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537:
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523:
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508:
494:
489:
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449:
444:
439:
433:
423:
409:
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375:
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320:
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308:
305:
302:
297:
294:
287:
284:
281:
278:
275:
272:
269:
264:
261:
255:
250:
245:
241:
238:
234:
225:
221:
217:
213:
203:
190:
184:
180:
177:
173:
170:
164:
161:
158:
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144:
141:
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119:
116:
113:
109:
105:
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96:
93:
89:
80:
76:
70:
66:
55:
53:
49:
45:
44:matrix theory
41:
37:
30:
21:
2228:
2160:Substitution
2046:graph theory
1543:Quaternionic
1533:Persymmetric
1412:
1261:
1257:
1232:
1227:Muir, Thomas
1200:
1196:
1190:
1171:
1167:
1157:
1140:
1136:
1104:
1100:
1090:
1073:
1069:
1040:real numbers
1036:ground field
1034:. When the
1026:
1012:
1005:
996:
992:
981:
977:
960:
956:
953:
943:
939:
805:
796:
792:
789:
715:
711:
679:
675:
671:
665:
661:
657:
653:eigenvectors
629:
625:
607:
603:
593:. Since the
589:
585:
581:
556:
552:
539:
535:
353:
349:
224:antidiagonal
215:
211:
209:denotes the
204:
81:
74:
68:
64:
61:
47:
33:
2135:Hamiltonian
2059:Biadjacency
1995:Correlation
1911:Householder
1861:Commutation
1598:Vandermonde
1593:Tridiagonal
1528:Permutation
1518:Nonnegative
1503:Matrix unit
1383:Bisymmetric
1174:: 207–218.
990:satisfying
696:eigenvalues
648:, then its
36:mathematics
2263:Categories
2035:Transition
2030:Stochastic
2000:Covariance
1982:statistics
1961:Symplectic
1956:Similarity
1785:Unimodular
1780:Orthogonal
1765:Involutory
1760:Invertible
1755:Projection
1751:Idempotent
1693:Convergent
1588:Triangular
1538:Polynomial
1483:Hessenberg
1453:Equivalent
1448:Elementary
1428:Copositive
1418:Conference
1378:Bidiagonal
1050:References
674:= −
646:eigenbasis
617:subalgebra
2216:Wronskian
2140:Irregular
2130:Gell-Mann
2079:Laplacian
2074:Incidence
2054:Adjacency
2025:Precision
1990:Centering
1896:Generator
1866:Confusion
1851:Circulant
1831:Augmented
1790:Unipotent
1770:Nilpotent
1746:Congruent
1723:Stieltjes
1698:Defective
1688:Companion
1659:Redheffer
1578:Symmetric
1573:Sylvester
1548:Signature
1478:Hermitian
1458:Frobenius
1368:Arrowhead
1348:Alternant
1296:MathWorld
1044:Hermitian
1028:Symmetric
908:…
895:∈
866:−
847:−
836:−
633:matrices.
510:Symmetric
312:≠
178:…
165:∈
136:−
117:−
2274:Matrices
2044:Used in
1980:Used in
1941:Rotation
1916:Jacobian
1876:Distance
1856:Cofactor
1841:Carleman
1821:Adjugate
1805:Weighing
1738:inverses
1734:products
1703:Definite
1634:Identity
1624:Exchange
1617:Constant
1583:Toeplitz
1468:Hadamard
1438:Diagonal
1229:(1960).
947:, where
615:forms a
564:for any
513:Toeplitz
361:Examples
2145:Overlap
2110:Density
2069:Edmonds
1946:Seifert
1906:Hessian
1871:Coxeter
1795:Unitary
1713:Hurwitz
1644:Of ones
1629:Hilbert
1563:Skyline
1508:Metzler
1498:Logical
1493:Integer
1403:Boolean
1335:classes
1278:2323222
1038:is the
1013:AK = KA
1002:integer
1000:for an
984:
975:(i.e.,
800:matrix
700:commute
678:
664:
623:of all
619:of the
72:matrix
2064:Degree
2005:Design
1936:Random
1926:Payoff
1921:Moment
1846:Cartan
1836:Bézout
1775:Normal
1649:Pascal
1639:Lehmer
1568:Sparse
1488:Hollow
1473:Hankel
1408:Cauchy
1333:Matrix
1276:
1245:
1008:> 1
683:where
52:matrix
2125:Gamma
2089:Tutte
1951:Shear
1664:Shift
1654:Pauli
1603:Walsh
1513:Moore
1393:Block
1274:JSTOR
702:with
545:field
50:is a
1931:Pick
1901:Gram
1669:Zero
1373:Band
1243:ISBN
560:and
533:are
529:and
46:, a
42:and
2020:Hat
1753:or
1736:or
1294:on
1266:doi
1205:doi
1176:doi
1172:377
1145:doi
1109:doi
1078:doi
942:= −
790:An
753:mod
690:If
669:or
636:If
601:of
599:set
590:ABJ
586:AJB
582:JAB
568:in
525:If
62:An
34:In
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1201:25
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1135:.
1123:^
1103:.
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1074:32
1072:.
1058:^
995:=
980:=
961:JA
959:=
957:AJ
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660:=
588:=
584:=
577:AB
562:cA
555:+
357:.
354:JA
352:=
350:AJ
214:×
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2150:S
1608:Z
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1268::
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1207::
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1151:.
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1117:.
1111::
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1084:.
1080::
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768:.
763:2
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672:x
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642:m
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630:n
626:n
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608:n
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495:.
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