677:
329:
532:
218:
537:
772:
Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator:
201:
When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in
743:. The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis (
401:
are respectively the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as
391:
121:
741:
527:
493:
349:
431:
699:
672:{\displaystyle {\begin{aligned}&&y'&=Sx'\\&\Rightarrow &Py&=SPx\\&\Rightarrow &y&=\left(P^{-1}SP\right)x=Tx\end{aligned}}}
906:
does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field;
914:
are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of
918:; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the
324:{\displaystyle S={\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}},}
24:
1010:. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar.
956:
either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries).
937:(the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of
211:
is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive
819:
1213:
1187:
1131:
185:
of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that
81:
894:. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really
354:
208:
898:; moreover their determination depends on being able to factor the minimal or characteristic polynomial of
798:
404:
1065:
states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.
887:
782:
812:
of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix
903:
704:
351:
is the angle of rotation. In the new coordinate system, the transformation would be written as
844:
809:
802:
793:
128:
1074:
875:
824:
777:
334:
20:
1123:
766:
498:
464:
168:
49:
8:
1118:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
1062:
840:
1116:
1084:
1018:
829:
684:
1208:
1183:
1156:
1127:
1079:
919:
64:
31:
1054:
835:
1089:
968:
879:
173:
136:
1152:
1038:
995:
895:
883:
38:
1202:
1058:
843:, which form a complete set of invariants for similarity of matrices over a
891:
788:
922:, over the ring of polynomials, of the matrix (with polynomial entries)
124:
449:
in terms of the simpler matrix, we use the change-of-basis matrix
1160:
959:
Similarity of matrices does not depend on the base field: if
882:. Not all matrices are diagonalizable, but at least over the
1194:(Similarity is discussed many places, starting at page 44.)
1002:. This is so because the rational canonical form over
854:, one is interested in finding a simple "normal form"
233:
707:
687:
535:
501:
467:
407:
357:
337:
221:
84:
16:
Equivalence under a change of basis (linear algebra)
1115:
1114:Beauregard, Raymond A.; Fraleigh, John B. (1973).
1113:
735:
693:
671:
521:
487:
425:
385:
343:
323:
115:
1061:is unitarily equivalent to some diagonal matrix.
1200:
866:then reduces to the study of the simpler matrix
1013:In the definition of similarity, if the matrix
785:, and attributes that can be derived from it:
1178:Horn, Roger A.; Johnson, Charles R. (1985).
902:(equivalently to find its eigenvalues). The
1177:
890:), every matrix is similar to a matrix in
832:, up to a permutation of the Jordan blocks
25:Similarity transformation (disambiguation)
1006:is also the rational canonical form over
681:Thus, the matrix in the original basis,
1146:
177:, and similar matrices are also called
1201:
941:itself; moreover it is not similar to
747:), perform the simple transformation (
171:, similarity is therefore the same as
1170:
751:), and change back to the old basis (
196:
850:Because of this, for a given matrix
445:are in the original basis. To write
123:Similar matrices represent the same
13:
998:they are similar as matrices over
14:
1225:
769:on the space of square matrices.
441:and the unknown transform matrix
1149:Matrix Methods: An Introduction
215:-axis, then it would simply be
181:; however, in a given subgroup
127:under two (possibly) different
1182:. Cambridge University Press.
1140:
1107:
609:
575:
1:
1095:
991:are similar as matrices over
760:
1100:
7:
1068:
116:{\displaystyle B=P^{-1}AP.}
10:
1230:
888:algebraically closed field
736:{\displaystyle T=P^{-1}SP}
29:
18:
1214:Equivalence (mathematics)
1147:Bronson, Richard (1970),
783:Characteristic polynomial
157:similarity transformation
810:Geometric multiplicities
803:algebraic multiplicities
30:Not to be confused with
904:rational canonical form
386:{\displaystyle y'=Sx',}
344:{\displaystyle \theta }
1037:can be chosen to be a
1017:can be chosen to be a
979:are two matrices over
963:is a field containing
878:if it is similar to a
845:principal ideal domain
737:
695:
673:
523:
489:
427:
387:
345:
325:
117:
1051:unitarily equivalent.
825:Frobenius normal form
738:
696:
674:
524:
522:{\displaystyle y'=Py}
490:
488:{\displaystyle x'=Px}
428:
426:{\displaystyle y=Tx,}
388:
346:
326:
118:
21:Similarity (geometry)
1155:, pp. 176–178,
1124:Houghton Mifflin Co.
1031:permutation-similar;
858:which is similar to
767:equivalence relation
705:
685:
533:
499:
465:
405:
355:
335:
219:
189:be chosen to lie in
169:general linear group
82:
19:For other uses, see
841:Elementary divisors
836:Index of nilpotence
63:if there exists an
1171:General references
1126:pp. 240–243.
1085:Matrix equivalence
1019:permutation matrix
830:Jordan normal form
820:Minimal polynomial
733:
691:
669:
667:
519:
485:
423:
383:
341:
321:
312:
197:Motivating example
113:
1080:Matrix congruence
920:Smith normal form
765:Similarity is an
694:{\displaystyle T}
142:A transformation
32:similarity matrix
1221:
1193:
1164:
1163:
1144:
1138:
1137:
1121:
1111:
1063:Specht's theorem
1057:says that every
1055:spectral theorem
955:
936:
756:
750:
746:
742:
740:
739:
734:
726:
725:
700:
698:
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678:
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675:
670:
668:
652:
648:
641:
640:
605:
571:
567:
549:
540:
539:
528:
526:
525:
520:
509:
494:
492:
491:
486:
475:
460:
456:
453:that transforms
452:
448:
444:
440:
436:
432:
430:
429:
424:
400:
396:
392:
390:
389:
384:
379:
365:
350:
348:
347:
342:
330:
328:
327:
322:
317:
316:
214:
209:axis of rotation
206:
192:
188:
184:
166:
154:
134:
122:
120:
119:
114:
103:
102:
77:
58:
54:
1229:
1228:
1224:
1223:
1222:
1220:
1219:
1218:
1199:
1198:
1197:
1190:
1180:Matrix Analysis
1173:
1168:
1167:
1145:
1141:
1134:
1112:
1108:
1103:
1098:
1090:Jacobi rotation
1075:Canonical forms
1071:
950:
942:
931:
923:
884:complex numbers
880:diagonal matrix
870:. For example,
763:
752:
748:
744:
718:
714:
706:
703:
702:
686:
683:
682:
666:
665:
633:
629:
628:
624:
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612:
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602:
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578:
569:
568:
560:
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542:
536:
534:
531:
530:
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442:
438:
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402:
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372:
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352:
336:
333:
332:
311:
310:
305:
300:
294:
293:
288:
277:
265:
264:
259:
245:
229:
228:
220:
217:
216:
212:
202:
199:
190:
186:
182:
164:
143:
137:change of basis
132:
95:
91:
83:
80:
79:
75:
56:
52:
35:
28:
17:
12:
11:
5:
1227:
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1153:Academic Press
1139:
1132:
1105:
1104:
1102:
1099:
1097:
1094:
1093:
1092:
1087:
1082:
1077:
1070:
1067:
1039:unitary matrix
996:if and only if
946:
927:
876:diagonalizable
862:—the study of
848:
847:
838:
833:
827:
822:
817:
807:
806:
805:
796:
791:
780:
762:
759:
732:
729:
724:
721:
717:
713:
710:
701:, is given by
690:
664:
661:
658:
655:
651:
647:
644:
639:
636:
632:
627:
623:
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582:
579:
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566:
563:
559:
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551:
548:
545:
541:
538:
518:
515:
512:
508:
505:
484:
481:
478:
474:
471:
433:where vectors
422:
419:
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413:
410:
382:
378:
375:
371:
368:
364:
361:
340:
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309:
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255:
252:
249:
246:
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241:
238:
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234:
232:
227:
224:
198:
195:
163:of the matrix
112:
109:
106:
101:
98:
94:
90:
87:
39:linear algebra
15:
9:
6:
4:
3:
2:
1226:
1215:
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1210:
1207:
1206:
1204:
1191:
1189:0-521-38632-2
1185:
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1175:
1162:
1158:
1154:
1150:
1143:
1135:
1133:0-395-14017-X
1129:
1125:
1120:
1119:
1110:
1106:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1072:
1066:
1064:
1060:
1059:normal matrix
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1011:
1009:
1005:
1001:
997:
994:
990:
986:
982:
978:
974:
970:
966:
962:
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935:
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926:
921:
917:
913:
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905:
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828:
826:
823:
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795:
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787:
786:
784:
781:
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776:
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768:
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727:
722:
719:
715:
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708:
688:
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637:
634:
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621:
619:
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583:
580:
573:
564:
561:
557:
554:
552:
546:
543:
516:
513:
510:
506:
503:
482:
479:
476:
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469:
420:
417:
414:
411:
408:
380:
376:
373:
369:
366:
362:
359:
338:
318:
313:
307:
302:
297:
290:
285:
282:
279:
274:
271:
268:
261:
256:
253:
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247:
242:
239:
236:
230:
225:
222:
210:
205:
194:
180:
176:
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162:
158:
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146:
140:
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126:
110:
107:
104:
99:
96:
92:
88:
85:
73:
69:
66:
62:
51:
48:
44:
40:
33:
26:
22:
1179:
1151:, New York:
1148:
1142:
1117:
1109:
1050:
1046:
1042:
1034:
1030:
1026:
1022:
1014:
1012:
1007:
1003:
999:
992:
988:
984:
980:
976:
972:
964:
960:
958:
952:
947:
943:
938:
933:
928:
924:
915:
911:
907:
899:
896:normal forms
871:
867:
863:
859:
855:
851:
849:
813:
801:, and their
771:
764:
753:
680:
203:
200:
178:
172:
160:
156:
155:is called a
151:
148:
144:
141:
71:
67:
60:
46:
42:
36:
892:Jordan form
799:Eigenvalues
789:Determinant
161:conjugation
59:are called
1203:Categories
1122:. Boston:
1096:References
874:is called
761:Properties
167:. In the
135:being the
125:linear map
78:such that
65:invertible
1101:Citations
720:−
635:−
610:⇒
576:⇒
339:θ
286:θ
283:
275:θ
272:
257:θ
254:
248:−
243:θ
240:
207:when the
179:conjugate
174:conjugacy
97:−
1209:Matrices
1161:70097490
1069:See also
969:subfield
886:(or any
565:′
547:′
507:′
473:′
377:′
363:′
139:matrix.
50:matrices
983:, then
131:, with
74:matrix
61:similar
1186:
1159:
1130:
971:, and
816:used).
393:where
331:where
41:, two
1041:then
1021:then
967:as a
794:Trace
129:bases
1184:ISBN
1157:LCCN
1128:ISBN
1053:The
1049:are
1045:and
1029:are
1025:and
987:and
975:and
910:and
778:Rank
495:and
457:and
437:and
397:and
70:-by-
55:and
45:-by-
23:and
1033:if
757:).
461:as
280:cos
269:sin
251:sin
237:cos
159:or
37:In
1205::
951:−
944:XI
932:−
925:XI
529::
399:y'
395:x'
193:.
152:AP
147:↦
1192:.
1136:.
1047:B
1043:A
1035:P
1027:B
1023:A
1015:P
1008:L
1004:K
1000:L
993:K
989:B
985:A
981:K
977:B
973:A
965:K
961:L
953:A
948:n
939:A
934:A
929:n
916:A
912:B
908:A
900:A
872:A
868:B
864:A
860:A
856:B
852:A
814:P
754:P
749:S
745:P
731:P
728:S
723:1
716:P
712:=
709:T
689:T
663:x
660:T
657:=
654:x
650:)
646:P
643:S
638:1
631:P
626:(
622:=
615:y
600:x
597:P
594:S
591:=
584:y
581:P
562:x
558:S
555:=
544:y
517:y
514:P
511:=
504:y
483:x
480:P
477:=
470:x
459:y
455:x
451:P
447:T
443:T
439:y
435:x
421:,
418:x
415:T
412:=
409:y
381:,
374:x
370:S
367:=
360:y
319:,
314:]
308:1
303:0
298:0
291:0
262:0
231:[
226:=
223:S
213:z
204:R
191:H
187:P
183:H
165:A
149:P
145:A
133:P
111:.
108:P
105:A
100:1
93:P
89:=
86:B
76:P
72:n
68:n
57:B
53:A
47:n
43:n
34:.
27:.
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