1324:
227:"A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares"
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415:
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1206:
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809:
489:
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114:
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917:
514:
401:
376:
120:: two matrices are congruent if and only if they represent the same bilinear form with respect to different
1116:
699:
504:
289:
121:
1062:
799:
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644:
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614:
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301:
165:. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.
91:
1277:
1251:
829:
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174:
139:
132:
47:
1328:
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380:
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179:
55:
305:
1334:
1302:
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1170:
1165:
1145:
1081:
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957:
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669:
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459:
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154:
87:
1032:
967:
947:
932:
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896:
794:
725:
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674:
559:
529:
98:
142:) rather than transpose, but this definition has not been adopted by most other authors.
1292:
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1201:
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277:
136:
110:
270:
1354:
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1155:
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1027:
1017:
1012:
937:
866:
740:
730:
659:
579:
564:
499:
349:
106:
32:
294:
1180:
1137:
1042:
755:
694:
604:
484:
117:
226:
17:
1022:
992:
760:
594:
464:
323:
203:
158:
128:
102:
28:
1073:
534:
162:
1307:
881:
1241:
393:
397:
161:
entries have the same numbers of positive, negative, and zero
97:
Matrix congruence arises when considering the effect of
1265:
1189:
1135:
1071:
905:
823:
769:
708:
432:
293:
269:
354:A survey of matrix theory and matrix inequalities
409:
8:
983:Fundamental (linear differential equation)
416:
402:
394:
1288:Matrix representation of conic sections
195:
7:
356:. Dover Publications. p. 81.
25:
328:An introduction to linear algebra
1322:
208:Finite dimensional vector spaces
1190:Used in science and engineering
131:defines congruence in terms of
63:over the same field such that
433:Explicitly constrained entries
1:
1207:Fundamental (computer vision)
973:Duplication and elimination
772:eigenvalues or eigenvectors
263:. van Nostrand. p. 80.
90:. Matrix congruence is an
1387:
906:With specific applications
535:Discrete Fourier Transform
153:states that two congruent
151:Sylvester's law of inertia
1371:Equivalence (mathematics)
1316:
1197:Cabibbo–Kobayashi–Maskawa
824:Satisfying conditions on
146:Congruence over the reals
555:Generalized permutation
377:Oxford University Press
352:; Minc, Henryk (1992).
225:Sylvester, J J (1852).
1329:Mathematics portal
234:Philosophical Magazine
86:where "T" denotes the
373:Undergraduate algebra
371:Norman, C.W. (1986).
259:; Weir, A.J. (1967).
92:equivalence relation
1278:Linear independence
525:Diagonally dominant
268:Hadley, G. (1961).
175:Congruence relation
140:inner product space
135:(with respect to a
133:conjugate transpose
54:if there exists an
1283:Matrix exponential
1273:Jordan normal form
1107:Fisher information
978:Euclidean distance
892:Totally unimodular
332:Dover Publications
185:Matrix equivalence
155:symmetric matrices
115:finite-dimensional
18:Congruent matrices
1348:
1347:
1340:Category:Matrices
1212:Fuzzy associative
1102:Doubly stochastic
810:Positive-definite
490:Block tridiagonal
296:Topics in algebra
180:Matrix similarity
56:invertible matrix
16:(Redirected from
1378:
1335:List of matrices
1327:
1326:
1303:Row echelon form
1247:State transition
1176:Seidel adjacency
1058:Totally positive
918:Alternating sign
515:Complex Hadamard
418:
411:
404:
395:
390:
367:
345:
319:
299:
285:
275:
264:
249:
248:
246:
245:
231:
222:
216:
215:
200:
88:matrix transpose
21:
1386:
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1351:
1350:
1349:
1344:
1321:
1312:
1261:
1185:
1131:
1067:
901:
819:
765:
704:
505:Centrosymmetric
428:
422:
387:
379:. p. 354.
370:
364:
348:
342:
334:. p. 182.
322:
316:
288:
267:
261:Linear geometry
257:Gruenberg, K.W.
255:
252:
243:
241:
229:
224:
223:
219:
204:Halmos, Paul R.
202:
201:
197:
193:
171:
148:
99:change of basis
33:square matrices
23:
22:
15:
12:
11:
5:
1384:
1382:
1374:
1373:
1368:
1363:
1361:Linear algebra
1353:
1352:
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1345:
1343:
1342:
1337:
1332:
1317:
1314:
1313:
1311:
1310:
1305:
1300:
1295:
1293:Perfect matrix
1290:
1285:
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1263:
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1260:
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1193:
1191:
1187:
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1184:
1183:
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1173:
1168:
1163:
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1148:
1142:
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1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1078:
1076:
1069:
1068:
1066:
1065:
1063:Transformation
1060:
1055:
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1040:
1035:
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1020:
1015:
1010:
1005:
1000:
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985:
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965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
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909:
907:
903:
902:
900:
899:
894:
889:
884:
879:
874:
869:
864:
859:
854:
849:
840:
834:
832:
821:
820:
818:
817:
812:
807:
802:
800:Diagonalizable
797:
792:
787:
782:
776:
774:
770:Conditions on
767:
766:
764:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
712:
710:
706:
705:
703:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
650:Skew-symmetric
647:
645:Skew-Hermitian
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
532:
527:
522:
517:
512:
507:
502:
497:
492:
487:
482:
480:Block-diagonal
477:
472:
467:
462:
457:
455:Anti-symmetric
452:
450:Anti-Hermitian
447:
442:
436:
434:
430:
429:
423:
421:
420:
413:
406:
398:
392:
391:
385:
368:
362:
350:Marcus, Marvin
346:
340:
320:
314:
290:Herstein, I.N.
286:
278:Addison-Wesley
272:Linear algebra
265:
251:
250:
217:
214:. p. 134.
194:
192:
189:
188:
187:
182:
177:
170:
167:
147:
144:
111:quadratic form
105:attached to a
84:
83:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1383:
1372:
1369:
1367:
1364:
1362:
1359:
1358:
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1341:
1338:
1336:
1333:
1331:
1330:
1325:
1319:
1318:
1315:
1309:
1306:
1304:
1301:
1299:
1298:Pseudoinverse
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1270:
1268:
1266:Related terms
1264:
1258:
1257:Z (chemistry)
1255:
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1200:
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1128:
1125:
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1120:
1118:
1115:
1113:
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1108:
1105:
1103:
1100:
1098:
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1021:
1019:
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1011:
1009:
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1004:
1001:
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994:
991:
989:
986:
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969:
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648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
615:Pentadiagonal
613:
611:
608:
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603:
601:
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593:
591:
588:
586:
583:
581:
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568:
566:
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561:
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551:
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541:
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531:
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511:
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491:
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486:
483:
481:
478:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
446:
445:Anti-diagonal
443:
441:
438:
437:
435:
431:
426:
419:
414:
412:
407:
405:
400:
399:
396:
388:
386:0-19-853248-2
382:
378:
374:
369:
365:
363:0-486-67102-X
359:
355:
351:
347:
343:
341:0-486-66434-1
337:
333:
329:
325:
321:
317:
315:0-471-02371-X
311:
307:
303:
298:
297:
291:
287:
283:
279:
274:
273:
266:
262:
258:
254:
253:
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228:
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209:
205:
199:
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186:
183:
181:
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172:
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166:
164:
160:
156:
152:
145:
143:
141:
138:
134:
130:
125:
123:
119:
116:
112:
108:
107:bilinear form
104:
100:
95:
93:
89:
82:
81:
76:
75:
71:
70:
66:
65:
64:
62:
61:
57:
53:
49:
45:
44:
39:
38:
34:
30:
19:
1320:
1252:Substitution
1138:graph theory
837:
635:Quaternionic
625:Persymmetric
372:
353:
327:
295:
271:
260:
242:. Retrieved
237:
233:
220:
212:van Nostrand
207:
198:
149:
126:
118:vector space
96:
85:
79:
78:
73:
72:
68:
67:
59:
58:
51:
42:
41:
36:
35:
26:
1227:Hamiltonian
1151:Biadjacency
1087:Correlation
1003:Householder
953:Commutation
690:Vandermonde
685:Tridiagonal
620:Permutation
610:Nonnegative
595:Matrix unit
475:Bisymmetric
163:eigenvalues
103:Gram matrix
50:are called
29:mathematics
1355:Categories
1127:Transition
1122:Stochastic
1092:Covariance
1074:statistics
1053:Symplectic
1048:Similarity
877:Unimodular
872:Orthogonal
857:Involutory
852:Invertible
847:Projection
843:Idempotent
785:Convergent
680:Triangular
630:Polynomial
575:Hessenberg
545:Equivalent
540:Elementary
520:Copositive
510:Conference
470:Bidiagonal
324:Mirsky, L.
304:. p.
280:. p.
244:2007-12-30
191:References
127:Note that
1308:Wronskian
1232:Irregular
1222:Gell-Mann
1171:Laplacian
1166:Incidence
1146:Adjacency
1117:Precision
1082:Centering
988:Generator
958:Confusion
943:Circulant
923:Augmented
882:Unipotent
862:Nilpotent
838:Congruent
815:Stieltjes
790:Defective
780:Companion
751:Redheffer
670:Symmetric
665:Sylvester
640:Signature
570:Hermitian
550:Frobenius
460:Arrowhead
440:Alternant
240:: 138–142
52:congruent
1366:Matrices
1136:Used in
1072:Used in
1033:Rotation
1008:Jacobian
968:Distance
948:Cofactor
933:Carleman
913:Adjugate
897:Weighing
830:inverses
826:products
795:Definite
726:Identity
716:Exchange
709:Constant
675:Toeplitz
560:Hadamard
530:Diagonal
326:(1990).
292:(1975).
206:(1958).
169:See also
1237:Overlap
1202:Density
1161:Edmonds
1038:Seifert
998:Hessian
963:Coxeter
887:Unitary
805:Hurwitz
736:Of ones
721:Hilbert
655:Skyline
600:Metzler
590:Logical
585:Integer
495:Boolean
427:classes
137:complex
101:on the
46:over a
1156:Degree
1097:Design
1028:Random
1018:Payoff
1013:Moment
938:Cartan
928:BĂ©zout
867:Normal
741:Pascal
731:Lehmer
660:Sparse
580:Hollow
565:Hankel
500:Cauchy
425:Matrix
383:
360:
338:
312:
129:Halmos
31:, two
1217:Gamma
1181:Tutte
1043:Shear
756:Shift
746:Pauli
695:Walsh
605:Moore
485:Block
302:Wiley
230:(PDF)
157:with
122:bases
113:on a
48:field
1023:Pick
993:Gram
761:Zero
465:Band
381:ISBN
358:ISBN
336:ISBN
310:ISBN
159:real
40:and
1112:Hat
845:or
828:or
306:352
282:253
109:or
27:In
1357::
375:.
330:.
308:.
300:.
276:.
238:IV
236:.
232:.
210:.
124:.
94:.
77:=
74:AP
1242:S
700:Z
417:e
410:t
403:v
389:.
366:.
344:.
318:.
284:.
247:.
80:B
69:P
60:P
43:B
37:A
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.