140:
which the action is as simple as it can get, namely by a scalar), and then similarity can be decided by comparing eigenvalues and their multiplicities. While in practice this is often a quite insightful approach, there are various drawbacks this has as a general method. First, it requires finding all eigenvalues, say as roots of the characteristic polynomial, but it may not be possible to give an explicit expression for them. Second, a complete set of eigenvalues might exist only in an extension of the field one is working over, and then one does not get a proof of similarity over the original field. Finally
2502:
640:
1938:
178:). Therefore, just having for both matrices some decomposition of the space into cyclic subspaces, and knowing the corresponding minimal polynomials, is not in itself sufficient to decide their similarity. An additional condition is imposed to ensure that for similar matrices one gets decompositions into cyclic subspaces that exactly match: in the list of associated minimal polynomials each one must divide the next (and the constant polynomial 1 is forbidden to exclude trivial cyclic subspaces of dimension 0). The resulting list of polynomials are called the
2062:
212:
1558:
2497:{\displaystyle \scriptstyle P={\begin{pmatrix}3&5&1&-1&0&0&-4&0\\4&4&0&-1&-1&-2&-3&-5\\8&5&0&-2&-5&-2&-11&-6\\0&9&0&-1&3&-2&0&0\\-1&-1&0&0&0&1&-1&4\\0&1&0&0&0&0&-1&1\\2&1&0&1&-1&0&2&-6\\-1&-2&0&0&1&-1&4&-2\end{pmatrix}}}
635:{\displaystyle \scriptstyle A={\begin{pmatrix}-1&3&-1&0&-2&0&0&-2\\-1&-1&1&1&-2&-1&0&-1\\-2&-6&4&3&-8&-4&-2&1\\-1&8&-3&-1&5&2&3&-3\\0&0&0&0&0&0&0&1\\0&0&0&0&-1&0&0&0\\1&0&0&0&2&0&0&0\\0&0&0&0&4&0&1&0\end{pmatrix}}.}
1933:{\displaystyle \scriptstyle C={\begin{pmatrix}0&1&0&0&0&0&0&0\\1&1&0&0&0&0&0&0\\0&0&0&0&0&0&0&-1\\0&0&1&0&0&0&0&-4\\0&0&0&1&0&0&0&-4\\0&0&0&0&1&0&0&2\\0&0&0&0&0&1&0&4\\0&0&0&0&0&0&1&0\end{pmatrix}}.}
3130:
3558:
151:
But obtaining such a fine decomposition is not necessary to just decide whether two matrices are similar. The rational canonical form is based on instead using a direct sum decomposition into stable subspaces that are as large as possible, while still allowing a very simple description of the action
139:
are similar, one approach is to try, for each of them, to decompose the vector space as far as possible into a direct sum of stable subspaces, and compare the respective actions on these subspaces. For instance if both are diagonalizable, then one can take the decomposition into eigenspaces (for
165:
of a monic polynomial; this polynomial (the minimal polynomial of the operator restricted to the subspace, which notion is analogous to that of the order of a cyclic subgroup) determines the action of the operator on the cyclic subspace up to isomorphism, and is independent of the choice of the
2846:
is then the minimal polynomial, which all the invariant factors therefore divide, and the product of the invariant factors gives the characteristic polynomial. Note that this implies that the minimal polynomial divides the characteristic polynomial (which is essentially the
156:
and all its images by repeated application of the linear operator associated to the matrix; such subspaces are called cyclic subspaces (by analogy with cyclic subgroups) and they are clearly stable under the linear operator. A basis of such a subspace is obtained by taking
3016:
173:
A direct sum decomposition into cyclic subspaces always exists, and finding one does not require factoring polynomials. However it is possible that cyclic subspaces do allow a decomposition as direct sum of smaller cyclic subspaces (essentially by the
931:. There always exist vectors such that the cyclic subspace that they generate has the same minimal polynomial as the operator has on the whole space; indeed most vectors will have this property, and in this case the first standard basis vector
2764:
3215:
2940:(if the characteristic polynomial splits into linear factors). For instance, the Frobenius normal form of a diagonal matrix with distinct diagonal entries is just the companion matrix of its characteristic polynomial.
890:
2800:
one then takes the unique monic generators of the respective ideals, and since the structure theorem ensures containment of every ideal in the preceding ideal, one obtains the divisibility conditions for the
2892: = 1), the Frobenius normal form is the companion matrix of the characteristic polynomial. As the rational canonical form is uniquely determined by the unique invariant factors associated to
100:(whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician
1542:
745:
3186:. In any generalized Jordan block, all entries immediately below the main diagonal are 1. A basis of the cyclic module giving rise to this form is obtained by choosing a generating vector
1237:
1150:
3125:{\displaystyle \scriptstyle {\begin{pmatrix}C&0&\cdots &0\\U&C&\cdots &0\\\vdots &\ddots &\ddots &\vdots \\0&\cdots &U&C\end{pmatrix}}}
2054:
1043:
2928:). On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the
2010:
999:
148:
might not be diagonalizable even over this larger field, in which case one must instead use a decomposition into generalized eigenspaces, and possibly into Jordan blocks.
2551:
1269:
1446:
1387:
1328:
929:
956:
186:-module defined by) the matrix, and two matrices are similar if and only if they have identical lists of invariant factors. The rational canonical form of a matrix
1407:
1348:
1063:
111:. Instead of decomposing into a minimum number of cyclic subspaces, the primary form decomposes into a maximum number of cyclic subspaces. It is also defined over
3553:{\displaystyle v,A(v),A^{2}(v),\ldots ,A^{d-1}(v),~P(A)(v),A(P(A)(v)),\ldots ,A^{d-1}(P(A)(v)),~P^{2}(A)(v),\ldots ,~P^{k-1}(A)(v),\ldots ,A^{d-1}(P^{k-1}(A)(v))}
1967:
1289:
123:. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.
190:
is obtained by expressing it on a basis adapted to a decomposition into cyclic subspaces whose associated minimal polynomials are the invariant factors of
2994:-modules (like is done for the Frobenius normal form above), where the characteristic polynomial of each summand is still a (generally smaller) power of
2951:, but that does reflect a possible factorization of the characteristic polynomial (or equivalently the minimal polynomial) into irreducible factors over
3664:
2916:
The
Frobenius normal form does not reflect any form of factorization of the characteristic polynomial, even if it does exist over the ground field
754:
2851:), and that every irreducible factor of the characteristic polynomial also divides the minimal polynomial (possibly with lower multiplicity).
3006:
in the diagonal blocks, corresponding to a particular choice of a basis for the cyclic modules. This generalized Jordan block is itself a
161:
and its successive images as long as they are linearly independent. The matrix of the linear operator with respect to such a basis is the
2967:. It is based on the fact that the vector space can be canonically decomposed into a direct sum of stable subspaces corresponding to the
3624:
649:
3604:
119:, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of
2943:
There is another way to define a normal form, that, like the
Frobenius normal form, is always defined over the same field
1451:
1065:. There exist complementary stable subspaces (of dimension 2) to this cyclic subspace, and the space generated by vectors
654:
3698:
3151:
is a matrix whose sole nonzero entry is a 1 in the upper right hand corner. For the case of a linear irreducible factor
116:
2955:, and which reduces to the Jordan normal form when this factorization only contains linear factors (corresponding to
1155:
1068:
2599:
748:
175:
107:
Some authors use the term rational canonical form for a somewhat different form that is more properly called the
101:
81:
3693:
2848:
2775:-module. The structure theorem provides a decomposition into cyclic factors, each of which is a quotient of
747:, so that the dimension of a subspace generated by the repeated images of a single vector is at most 6. The
2976:
2637:
2015:
1004:
61:
76:). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix
2814:
68:. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for
50:
1972:
1552:
is the block diagonal matrix with the corresponding companion matrices as diagonal blocks, namely
961:
2937:
2818:
2704:
2511:
1242:
1412:
1353:
1294:
895:
3673:
3600:
3583:
2826:
2924:
is replaced by a different field (as long as it contains the entries of the original matrix
2864:
2700:
2587:
179:
162:
934:
3652:
2929:
2565:
1392:
1333:
1048:
93:
3002:
corresponding to such a decomposition into cyclic modules, with a particular form called
2981:
1946:
2990:. These summands can be further decomposed, non-canonically, as a direct sum of cyclic
2986:), where the characteristic polynomial of each summand is a power of the corresponding
1274:
57:
31:
17:
2888:. When the minimal polynomial is identical to the characteristic polynomial (the case
194:; two matrices are similar if and only if they have the same rational canonical form.
3687:
43:
3183:
3007:
2999:
2568:
3678:
1409:(and it is easily checked that it does), and we have found the invariant factors
2897:
2780:
2956:
2765:
structure theorem for finitely generated modules over a principal ideal domain
152:
on each of them. These subspaces must be generated by a single nonzero vector
92:. Since this form can be found without any operations that might change when
1045:
are linearly independent and span a cyclic subspace with minimal polynomial
885:{\displaystyle \chi =X^{8}-X^{7}-5X^{6}+2X^{5}+10X^{4}+2X^{3}-7X^{2}-5X-1}
2779:
by a proper ideal; the zero ideal cannot be present since the resulting
2908:
are similar if and only if they have the same rational canonical form.
2880:, and the block diagonal matrix formed from these blocks yields the
115:, but has somewhat different properties: finding the form requires
3599:. 2nd Edition, John Wiley & Sons. pp. 442, 446, 452-458.
1271:, so the complementary subspace is a cyclic subspace generated by
3619:
Phani
Bhushan Bhattacharya, Surender Kumar Jain, S. R. Nagpaul,
1943:
A basis on which this form is attained is formed by the vectors
2837:
as given above must be used instead. The last of these factors
1350:
is the minimal polynomial of the whole space, it is clear that
892:, which is a multiple of the minimal polynomial by a factor
72:(i.e., spanned by some vector and its repeated images under
2912:
A rational normal form generalizing the Jordan normal form
88:
if and only if it has the same rational canonical form as
3139:
is the companion matrix of the irreducible polynomial
3026:
3020:
2078:
2066:
1574:
1562:
228:
216:
3218:
3203:
where the minimal polynomial of the cyclic module is
3019:
2514:
2065:
2018:
1975:
1949:
1561:
1454:
1415:
1395:
1356:
1336:
1297:
1277:
1245:
1158:
1071:
1051:
1007:
964:
937:
898:
757:
657:
215:
1537:{\displaystyle \mu =X^{6}-4X^{4}-2X^{3}+4X^{2}+4X+1}
740:{\displaystyle \mu =X^{6}-4X^{4}-2X^{3}+4X^{2}+4X+1}
131:
When trying to find out whether two square matrices
64:
obtained by conjugation by invertible matrices over
2975:of the characteristic polynomial (as stated by the
3552:
3124:
2616:be a finite-dimensional vector space over a field
2545:
2496:
2048:
2004:
1961:
1932:
1536:
1440:
1401:
1381:
1342:
1322:
1283:
1263:
1231:
1144:
1057:
1037:
993:
950:
923:
884:
739:
634:
3637:Les maths en tête, Mathématiques pour M', Algèbre
2896:, and these invariant factors are independent of
2748:"; with these conditions the list of polynomials
3674:An Algorithm for the Frobenius Normal Form (pdf)
3165:, these blocks are reduced to single entries
1232:{\displaystyle w=(5,4,5,9,-1,1,1,-2)^{\top }}
1145:{\displaystyle v=(3,4,8,0,-1,0,2,-1)^{\top }}
8:
2998:. The primary rational canonical form is a
2791:is finite-dimensional. For the polynomials
3679:A rational canonical form Algorithm (pdf)
3517:
3498:
3455:
3415:
3366:
3272:
3244:
3217:
3021:
3018:
2936:is diagonalizable) or more generally the
2920:. This implies that it is invariant when
2602:and minimal polynomial are both equal to
2531:
2513:
2073:
2064:
2017:
1993:
1980:
1974:
1948:
1569:
1560:
1513:
1497:
1481:
1465:
1453:
1420:
1414:
1394:
1361:
1355:
1335:
1302:
1296:
1276:
1244:
1223:
1157:
1136:
1070:
1050:
1006:
982:
969:
963:
942:
936:
903:
897:
861:
845:
829:
813:
797:
781:
768:
756:
716:
700:
684:
668:
656:
223:
214:
3612:
27:Canonical form of matrices over a field
3595:David S. Dummit and Richard M. Foote.
2900:, it follows that two square matrices
2813:Given an arbitrary square matrix, the
1548:. Then the rational canonical form of
202:Consider the following matrix A, over
3669:) Algorithm for Frobenius Normal Form
2959:). This form is sometimes called the
7:
2703:of positive degree (so they are non-
3653:Rational Canonical Form (Mathworld)
1224:
1137:
25:
2783:would be infinite-dimensional as
2056:; explicitly this means that for
3192:(one that is not annihilated by
2817:used in the construction of the
2965:primary rational canonical form
2740:where "a | b" is notation for "
2049:{\displaystyle k=0,1,\ldots ,5}
1239:is an example. In fact one has
1038:{\displaystyle k=0,1,\ldots ,5}
109:primary rational canonical form
3639:, 1998, Ellipses, Th. 1 p. 173
3547:
3544:
3538:
3535:
3529:
3510:
3482:
3476:
3473:
3467:
3436:
3430:
3427:
3421:
3402:
3399:
3393:
3390:
3384:
3378:
3350:
3347:
3341:
3338:
3332:
3326:
3317:
3311:
3308:
3302:
3290:
3284:
3256:
3250:
3234:
3228:
3182:and, one finds a (transposed)
2961:generalized Jordan normal form
2586:, there is associated to it a
1999:
1986:
1220:
1165:
1133:
1078:
988:
975:
1:
2711:) that satisfy the relations
2005:{\displaystyle A^{k}(e_{1})}
1291:; it has minimal polynomial
994:{\displaystyle A^{k}(e_{1})}
117:factorization of polynomials
2546:{\displaystyle A=PCP^{-1}.}
3715:
2854:For each invariant factor
1264:{\displaystyle A\cdot v=w}
2600:characteristic polynomial
1441:{\displaystyle X^{2}-X-1}
1382:{\displaystyle X^{2}-X-1}
1323:{\displaystyle X^{2}-X-1}
924:{\displaystyle X^{2}-X-1}
749:characteristic polynomial
176:Chinese remainder theorem
170:generating the subspace.
102:Ferdinand Georg Frobenius
3004:generalized Jordan block
3209:), and taking as basis
2882:rational canonical form
2849:Cayley-Hamilton theorem
2556:General case and theory
40:rational canonical form
18:Rational canonical form
3621:Basic abstract algebra
3554:
3126:
2547:
2498:
2050:
2006:
1963:
1934:
1538:
1442:
1403:
1383:
1344:
1324:
1285:
1265:
1233:
1146:
1059:
1039:
995:
952:
925:
886:
741:
636:
3555:
3127:
3000:block diagonal matrix
2624:a square matrix over
2578:. Given a polynomial
2548:
2499:
2051:
2007:
1964:
1935:
1539:
1443:
1404:
1384:
1345:
1325:
1286:
1266:
1234:
1147:
1060:
1040:
996:
958:does so: the vectors
953:
951:{\displaystyle e_{1}}
926:
887:
742:
637:
36:Frobenius normal form
3216:
3017:
2971:irreducible factors
2810:. See for details.
2787:vector space, while
2652:-module isomorphism
2512:
2063:
2016:
1973:
1947:
1559:
1452:
1413:
1402:{\displaystyle \mu }
1393:
1354:
1343:{\displaystyle \mu }
1334:
1295:
1275:
1243:
1156:
1069:
1058:{\displaystyle \mu }
1049:
1005:
962:
935:
896:
755:
655:
213:
3699:Matrix normal forms
2815:elementary divisors
2771:, viewing it as an
2699:may be taken to be
2640:with the action of
1969:above, followed by
1962:{\displaystyle v,w}
3550:
3122:
3121:
3115:
2938:Jordan normal form
2821:do not exist over
2819:Jordan normal form
2543:
2494:
2493:
2487:
2046:
2002:
1959:
1930:
1929:
1920:
1534:
1438:
1399:
1379:
1340:
1320:
1281:
1261:
1229:
1142:
1055:
1035:
991:
948:
921:
882:
737:
650:minimal polynomial
632:
631:
622:
49:with entries in a
3635:Xavier Gourdon,
3584:Smith normal form
3450:
3410:
3298:
2827:invariant factors
2701:monic polynomials
2560:Fix a base field
1284:{\displaystyle v}
180:invariant factors
16:(Redirected from
3706:
3640:
3633:
3627:
3617:
3597:Abstract Algebra
3573:
3559:
3557:
3556:
3551:
3528:
3527:
3509:
3508:
3466:
3465:
3448:
3420:
3419:
3408:
3377:
3376:
3296:
3283:
3282:
3249:
3248:
3208:
3202:
3191:
3181:
3174:
3164:
3150:
3144:
3131:
3129:
3128:
3123:
3120:
3119:
2985:
2977:lemme des noyaux
2865:companion matrix
2588:companion matrix
2552:
2550:
2549:
2544:
2539:
2538:
2503:
2501:
2500:
2495:
2492:
2491:
2055:
2053:
2052:
2047:
2011:
2009:
2008:
2003:
1998:
1997:
1985:
1984:
1968:
1966:
1965:
1960:
1939:
1937:
1936:
1931:
1925:
1924:
1543:
1541:
1540:
1535:
1518:
1517:
1502:
1501:
1486:
1485:
1470:
1469:
1447:
1445:
1444:
1439:
1425:
1424:
1408:
1406:
1405:
1400:
1388:
1386:
1385:
1380:
1366:
1365:
1349:
1347:
1346:
1341:
1329:
1327:
1326:
1321:
1307:
1306:
1290:
1288:
1287:
1282:
1270:
1268:
1267:
1262:
1238:
1236:
1235:
1230:
1228:
1227:
1151:
1149:
1148:
1143:
1141:
1140:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1000:
998:
997:
992:
987:
986:
974:
973:
957:
955:
954:
949:
947:
946:
930:
928:
927:
922:
908:
907:
891:
889:
888:
883:
866:
865:
850:
849:
834:
833:
818:
817:
802:
801:
786:
785:
773:
772:
746:
744:
743:
738:
721:
720:
705:
704:
689:
688:
673:
672:
641:
639:
638:
633:
627:
626:
163:companion matrix
21:
3714:
3713:
3709:
3708:
3707:
3705:
3704:
3703:
3684:
3683:
3661:
3649:
3644:
3643:
3634:
3630:
3623:, Theorem 5.4,
3618:
3614:
3592:
3580:
3564:
3513:
3494:
3451:
3411:
3362:
3268:
3240:
3214:
3213:
3204:
3193:
3187:
3176:
3166:
3152:
3146:
3140:
3114:
3113:
3108:
3103:
3098:
3092:
3091:
3086:
3081:
3076:
3070:
3069:
3064:
3059:
3054:
3048:
3047:
3042:
3037:
3032:
3022:
3015:
3014:
2979:
2914:
2879:
2878:
2862:
2845:
2836:
2809:
2799:
2761:Sketch of Proof
2756:
2736:
2727:
2720:
2694:
2682:
2669:
2597:
2558:
2527:
2510:
2509:
2486:
2485:
2477:
2472:
2464:
2459:
2454:
2449:
2441:
2432:
2431:
2423:
2418:
2413:
2405:
2400:
2395:
2390:
2384:
2383:
2378:
2370:
2365:
2360:
2355:
2350:
2345:
2339:
2338:
2333:
2325:
2320:
2315:
2310:
2305:
2297:
2288:
2287:
2282:
2277:
2269:
2264:
2256:
2251:
2246:
2240:
2239:
2231:
2223:
2215:
2207:
2199:
2194:
2189:
2183:
2182:
2174:
2166:
2158:
2150:
2142:
2137:
2132:
2126:
2125:
2120:
2112:
2107:
2102:
2094:
2089:
2084:
2074:
2061:
2060:
2014:
2013:
1989:
1976:
1971:
1970:
1945:
1944:
1919:
1918:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1877:
1876:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1835:
1834:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1793:
1792:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1748:
1747:
1739:
1734:
1729:
1724:
1719:
1714:
1709:
1703:
1702:
1694:
1689:
1684:
1679:
1674:
1669:
1664:
1658:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1616:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1570:
1557:
1556:
1509:
1493:
1477:
1461:
1450:
1449:
1416:
1411:
1410:
1391:
1390:
1357:
1352:
1351:
1332:
1331:
1298:
1293:
1292:
1273:
1272:
1241:
1240:
1219:
1154:
1153:
1132:
1067:
1066:
1047:
1046:
1003:
1002:
978:
965:
960:
959:
938:
933:
932:
899:
894:
893:
857:
841:
825:
809:
793:
777:
764:
753:
752:
712:
696:
680:
664:
653:
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129:
28:
23:
22:
15:
12:
11:
5:
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3696:
3694:Linear algebra
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3648:
3647:External links
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2863:one takes its
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2752:
2738:
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2732:
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2718:
2690:
2684:
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2678:
2667:
2632:(viewed as an
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2001:
1996:
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1983:
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199:
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128:
125:
58:canonical form
32:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3711:
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3697:
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3672:
3670:
3668:
3663:
3662:
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3654:
3651:
3650:
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3638:
3632:
3629:
3626:
3622:
3616:
3613:
3606:
3605:0-471-36857-1
3602:
3598:
3594:
3593:
3589:
3585:
3582:
3581:
3577:
3575:
3571:
3567:
3541:
3532:
3524:
3521:
3518:
3514:
3505:
3502:
3499:
3495:
3491:
3488:
3485:
3479:
3470:
3462:
3459:
3456:
3452:
3445:
3442:
3439:
3433:
3424:
3416:
3412:
3405:
3396:
3387:
3381:
3373:
3370:
3367:
3363:
3359:
3356:
3353:
3344:
3335:
3329:
3323:
3320:
3314:
3305:
3299:
3293:
3287:
3279:
3276:
3273:
3269:
3265:
3262:
3259:
3253:
3245:
3241:
3237:
3231:
3225:
3222:
3219:
3212:
3211:
3210:
3207:
3200:
3196:
3190:
3185:
3179:
3173:
3169:
3163:
3159:
3155:
3149:
3143:
3138:
3116:
3110:
3105:
3100:
3095:
3088:
3083:
3078:
3073:
3066:
3061:
3056:
3051:
3044:
3039:
3034:
3029:
3023:
3013:
3012:
3011:
3009:
3005:
3001:
2997:
2993:
2989:
2983:
2978:
2974:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2941:
2939:
2935:
2931:
2930:diagonal form
2927:
2923:
2919:
2911:
2909:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2877:
2873:
2869:
2866:
2861:
2857:
2852:
2850:
2844:
2840:
2835:
2831:
2828:
2824:
2820:
2816:
2811:
2808:
2804:
2798:
2794:
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2755:
2751:
2747:
2743:
2735:
2731:
2724:
2717:
2714:
2713:
2712:
2710:
2706:
2702:
2698:
2693:
2689:
2681:
2677:
2673:
2666:
2662:
2658:
2655:
2654:
2653:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2605:
2601:
2596:
2592:
2589:
2585:
2581:
2577:
2573:
2570:
2567:
2564:and a finite-
2563:
2555:
2553:
2540:
2535:
2532:
2528:
2524:
2521:
2518:
2515:
2488:
2482:
2479:
2474:
2469:
2466:
2461:
2456:
2451:
2446:
2443:
2438:
2435:
2428:
2425:
2420:
2415:
2410:
2407:
2402:
2397:
2392:
2387:
2380:
2375:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2335:
2330:
2327:
2322:
2317:
2312:
2307:
2302:
2299:
2294:
2291:
2284:
2279:
2274:
2271:
2266:
2261:
2258:
2253:
2248:
2243:
2236:
2233:
2228:
2225:
2220:
2217:
2212:
2209:
2204:
2201:
2196:
2191:
2186:
2179:
2176:
2171:
2168:
2163:
2160:
2155:
2152:
2147:
2144:
2139:
2134:
2129:
2122:
2117:
2114:
2109:
2104:
2099:
2096:
2091:
2086:
2081:
2075:
2070:
2067:
2059:
2058:
2057:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
1994:
1990:
1981:
1977:
1956:
1953:
1950:
1926:
1921:
1915:
1910:
1905:
1900:
1895:
1890:
1885:
1880:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1838:
1831:
1826:
1821:
1816:
1811:
1806:
1801:
1796:
1789:
1786:
1781:
1776:
1771:
1766:
1761:
1756:
1751:
1744:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1699:
1696:
1691:
1686:
1681:
1676:
1671:
1666:
1661:
1654:
1649:
1644:
1639:
1634:
1629:
1624:
1619:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1571:
1566:
1563:
1555:
1554:
1553:
1551:
1547:
1531:
1528:
1525:
1522:
1519:
1514:
1510:
1506:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1471:
1466:
1462:
1458:
1455:
1435:
1432:
1429:
1426:
1421:
1417:
1396:
1376:
1373:
1370:
1367:
1362:
1358:
1337:
1317:
1314:
1311:
1308:
1303:
1299:
1278:
1258:
1255:
1252:
1249:
1246:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1162:
1159:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1075:
1072:
1052:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
983:
979:
970:
966:
943:
939:
918:
915:
912:
909:
904:
900:
879:
876:
873:
870:
867:
862:
858:
854:
851:
846:
842:
838:
835:
830:
826:
822:
819:
814:
810:
806:
803:
798:
794:
790:
787:
782:
778:
774:
769:
765:
761:
758:
750:
734:
731:
728:
725:
722:
717:
713:
709:
706:
701:
697:
693:
690:
685:
681:
677:
674:
669:
665:
661:
658:
651:
647:
628:
623:
617:
612:
607:
602:
597:
592:
587:
582:
575:
570:
565:
560:
555:
550:
545:
540:
533:
528:
523:
518:
515:
510:
505:
500:
495:
488:
483:
478:
473:
468:
463:
458:
453:
446:
443:
438:
433:
428:
423:
420:
415:
412:
407:
402:
399:
392:
387:
384:
379:
376:
371:
368:
363:
358:
353:
350:
345:
342:
335:
332:
327:
322:
319:
314:
311:
306:
301:
296:
293:
288:
285:
278:
275:
270:
265:
260:
257:
252:
247:
244:
239:
234:
231:
225:
220:
217:
209:
208:
207:
205:
197:
195:
193:
189:
185:
181:
177:
171:
169:
164:
160:
155:
149:
147:
143:
138:
134:
126:
124:
122:
118:
114:
110:
105:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
52:
48:
45:
44:square matrix
41:
37:
33:
19:
3666:
3636:
3631:
3620:
3615:
3596:
3569:
3565:
3562:
3205:
3198:
3194:
3188:
3184:Jordan block
3177:
3171:
3167:
3161:
3157:
3153:
3147:
3141:
3136:
3134:
3010:of the form
3008:block matrix
3003:
2995:
2991:
2987:
2972:
2968:
2964:
2960:
2952:
2948:
2944:
2942:
2933:
2925:
2921:
2917:
2915:
2905:
2901:
2893:
2889:
2885:
2881:
2875:
2871:
2867:
2859:
2855:
2853:
2842:
2838:
2833:
2829:
2822:
2812:
2806:
2802:
2796:
2792:
2788:
2784:
2776:
2772:
2768:
2763:: Apply the
2760:
2759:
2753:
2749:
2745:
2741:
2739:
2733:
2729:
2722:
2715:
2708:
2696:
2691:
2687:
2685:
2679:
2675:
2671:
2664:
2660:
2656:
2649:
2645:
2641:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2608:
2603:
2594:
2590:
2583:
2579:
2575:
2571:
2569:vector space
2561:
2559:
2507:
1942:
1549:
1545:
1389:must divide
645:
644:
203:
201:
191:
187:
183:
172:
167:
158:
153:
150:
145:
141:
136:
132:
130:
120:
112:
108:
106:
97:
89:
85:
77:
73:
69:
65:
53:
46:
39:
35:
29:
2980: [
2957:eigenvalues
2781:free module
2757:is unique.
2648:) admits a
2566:dimensional
3688:Categories
3659:Algorithms
3590:References
2686:where the
127:Motivation
96:the field
3522:−
3503:−
3489:…
3460:−
3443:…
3371:−
3357:…
3277:−
3263:…
3101:⋯
3089:⋮
3084:⋱
3079:⋱
3074:⋮
3062:⋯
3040:⋯
2825:, so the
2644:given by
2533:−
2480:−
2467:−
2444:−
2436:−
2426:−
2408:−
2373:−
2328:−
2300:−
2292:−
2272:−
2259:−
2234:−
2226:−
2218:−
2210:−
2202:−
2177:−
2169:−
2161:−
2153:−
2145:−
2115:−
2097:−
2038:…
1787:−
1742:−
1697:−
1488:−
1472:−
1456:μ
1433:−
1427:−
1397:μ
1374:−
1368:−
1338:μ
1315:−
1309:−
1250:⋅
1225:⊤
1214:−
1193:−
1138:⊤
1127:−
1106:−
1053:μ
1027:…
916:−
910:−
877:−
868:−
852:−
788:−
775:−
759:χ
691:−
675:−
659:μ
516:−
444:−
421:−
413:−
400:−
385:−
377:−
369:−
351:−
343:−
333:−
320:−
312:−
294:−
286:−
276:−
258:−
245:−
232:−
94:extending
3578:See also
2969:distinct
2744:divides
2508:one has
1330:. Since
182:of (the
62:matrices
2628:. Then
2610:Theorem
198:Example
166:vector
82:similar
3603:
3568:= deg(
3563:where
3449:
3409:
3297:
3145:, and
3135:where
2728:| … |
2670:⊕ … ⊕
2638:module
2620:, and
2612:: Let
2598:whose
34:, the
3665:An O(
3625:p.423
2984:]
2963:, or
2898:basis
2705:units
2574:over
56:is a
51:field
42:of a
3601:ISBN
3175:and
2932:(if
2904:and
2012:for
1448:and
1152:and
1001:for
648:has
144:and
135:and
80:is
60:for
38:or
3180:= 1
2947:as
2884:of
2767:to
2707:in
1544:of
751:is
84:to
30:In
3690::
3574:.
3170:=
3160:−
3156:=
2982:fr
2721:|
2695:∈
2659:≅
2606:.
2582:∈
2229:11
823:10
206::
104:.
3667:n
3607:.
3572:)
3570:P
3566:d
3548:)
3545:)
3542:v
3539:(
3536:)
3533:A
3530:(
3525:1
3519:k
3515:P
3511:(
3506:1
3500:d
3496:A
3492:,
3486:,
3483:)
3480:v
3477:(
3474:)
3471:A
3468:(
3463:1
3457:k
3453:P
3446:,
3440:,
3437:)
3434:v
3431:(
3428:)
3425:A
3422:(
3417:2
3413:P
3406:,
3403:)
3400:)
3397:v
3394:(
3391:)
3388:A
3385:(
3382:P
3379:(
3374:1
3368:d
3364:A
3360:,
3354:,
3351:)
3348:)
3345:v
3342:(
3339:)
3336:A
3333:(
3330:P
3327:(
3324:A
3321:,
3318:)
3315:v
3312:(
3309:)
3306:A
3303:(
3300:P
3294:,
3291:)
3288:v
3285:(
3280:1
3274:d
3270:A
3266:,
3260:,
3257:)
3254:v
3251:(
3246:2
3242:A
3238:,
3235:)
3232:v
3229:(
3226:A
3223:,
3220:v
3206:P
3201:)
3199:A
3197:(
3195:P
3189:v
3178:U
3172:λ
3168:C
3162:λ
3158:x
3154:P
3148:U
3142:P
3137:C
3117:)
3111:C
3106:U
3096:0
3067:0
3057:C
3052:U
3045:0
3035:0
3030:C
3024:(
2996:P
2992:F
2988:P
2973:P
2953:F
2949:A
2945:F
2934:A
2926:A
2922:F
2918:F
2906:B
2902:A
2894:A
2890:k
2886:A
2876:i
2872:f
2868:C
2860:i
2856:f
2843:k
2839:f
2834:i
2830:f
2823:F
2807:i
2803:f
2797:i
2793:f
2789:V
2785:F
2777:F
2773:F
2769:V
2754:i
2750:f
2746:b
2742:a
2734:k
2730:f
2726:2
2723:f
2719:1
2716:f
2709:F
2697:F
2692:i
2688:f
2680:k
2676:f
2674:/
2672:F
2668:1
2665:f
2663:/
2661:F
2657:V
2650:F
2646:A
2642:X
2636:-
2634:F
2630:V
2626:F
2622:A
2618:F
2614:V
2604:P
2595:P
2591:C
2584:F
2580:P
2576:F
2572:V
2562:F
2541:.
2536:1
2529:P
2525:C
2522:P
2519:=
2516:A
2504:,
2489:)
2483:2
2475:4
2470:1
2462:1
2457:0
2452:0
2447:2
2439:1
2429:6
2421:2
2416:0
2411:1
2403:1
2398:0
2393:1
2388:2
2381:1
2376:1
2368:0
2363:0
2358:0
2353:0
2348:1
2343:0
2336:4
2331:1
2323:1
2318:0
2313:0
2308:0
2303:1
2295:1
2285:0
2280:0
2275:2
2267:3
2262:1
2254:0
2249:9
2244:0
2237:6
2221:2
2213:5
2205:2
2197:0
2192:5
2187:8
2180:5
2172:3
2164:2
2156:1
2148:1
2140:0
2135:4
2130:4
2123:0
2118:4
2110:0
2105:0
2100:1
2092:1
2087:5
2082:3
2076:(
2071:=
2068:P
2044:5
2041:,
2035:,
2032:1
2029:,
2026:0
2023:=
2020:k
2000:)
1995:1
1991:e
1987:(
1982:k
1978:A
1957:w
1954:,
1951:v
1927:.
1922:)
1916:0
1911:1
1906:0
1901:0
1896:0
1891:0
1886:0
1881:0
1874:4
1869:0
1864:1
1859:0
1854:0
1849:0
1844:0
1839:0
1832:2
1827:0
1822:0
1817:1
1812:0
1807:0
1802:0
1797:0
1790:4
1782:0
1777:0
1772:0
1767:1
1762:0
1757:0
1752:0
1745:4
1737:0
1732:0
1727:0
1722:0
1717:1
1712:0
1707:0
1700:1
1692:0
1687:0
1682:0
1677:0
1672:0
1667:0
1662:0
1655:0
1650:0
1645:0
1640:0
1635:0
1630:0
1625:1
1620:1
1613:0
1608:0
1603:0
1598:0
1593:0
1588:0
1583:1
1578:0
1572:(
1567:=
1564:C
1550:A
1546:A
1532:1
1529:+
1526:X
1523:4
1520:+
1515:2
1511:X
1507:4
1504:+
1499:3
1495:X
1491:2
1483:4
1479:X
1475:4
1467:6
1463:X
1459:=
1436:1
1430:X
1422:2
1418:X
1377:1
1371:X
1363:2
1359:X
1318:1
1312:X
1304:2
1300:X
1279:v
1259:w
1256:=
1253:v
1247:A
1221:)
1217:2
1211:,
1208:1
1205:,
1202:1
1199:,
1196:1
1190:,
1187:9
1184:,
1181:5
1178:,
1175:4
1172:,
1169:5
1166:(
1163:=
1160:w
1134:)
1130:1
1124:,
1121:2
1118:,
1115:0
1112:,
1109:1
1103:,
1100:0
1097:,
1094:8
1091:,
1088:4
1085:,
1082:3
1079:(
1076:=
1073:v
1033:5
1030:,
1024:,
1021:1
1018:,
1015:0
1012:=
1009:k
989:)
984:1
980:e
976:(
971:k
967:A
944:1
940:e
919:1
913:X
905:2
901:X
880:1
874:X
871:5
863:2
859:X
855:7
847:3
843:X
839:2
836:+
831:4
827:X
820:+
815:5
811:X
807:2
804:+
799:6
795:X
791:5
783:7
779:X
770:8
766:X
762:=
735:1
732:+
729:X
726:4
723:+
718:2
714:X
710:4
707:+
702:3
698:X
694:2
686:4
682:X
678:4
670:6
666:X
662:=
646:A
629:.
624:)
618:0
613:1
608:0
603:4
598:0
593:0
588:0
583:0
576:0
571:0
566:0
561:2
556:0
551:0
546:0
541:1
534:0
529:0
524:0
519:1
511:0
506:0
501:0
496:0
489:1
484:0
479:0
474:0
469:0
464:0
459:0
454:0
447:3
439:3
434:2
429:5
424:1
416:3
408:8
403:1
393:1
388:2
380:4
372:8
364:3
359:4
354:6
346:2
336:1
328:0
323:1
315:2
307:1
302:1
297:1
289:1
279:2
271:0
266:0
261:2
253:0
248:1
240:3
235:1
226:(
221:=
218:A
204:Q
192:A
188:A
184:K
168:v
159:v
154:v
146:B
142:A
137:B
133:A
121:F
113:F
98:F
90:A
86:A
78:B
74:A
70:A
66:F
54:F
47:A
20:)
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