线性代数第21讲 相似矩阵

1、一、相似矩阵的概念二、相似矩阵的性质三、矩阵的相似对角化,1BAPP nBA,PBAAAPP1 APABn);)()1(111BPPAPPABPP ,)()2(111BPlPAPkPPlBkAP lk,ABPAB,1513 A,2004 B,5111 P,6/16/16/16/5 1P 511115136/16/16/16/5 APP1 2004 B ,1BAPP nABAB ABP PEPAPPEB)(11 PEAPPEAP 11)(,EA ABAB ,241153 EA ,242004 EB,41 , 22 ABAB ,P,1BAPP 111)( APPB1 A1 B1111)( PAP,

2、11PAP ABn,BA AB.)(2E ABA,P.111EPPEPPAPP ABBBBA,1001 A.1011 B,1APPB AB0 xA0 0 BxP1 ,0 xAx ,1APPB ,1 PBPA,01xxPBP 1 P)()(0111xPxPBPP )()(101xPxPB xP1 B0 nAn AP ,1 APP ,21npppP PAP n 21 nnppppppA,2121 ,21 n P,0 P), 2, 1(nipi nppp,21Anppp,21An,21n ), 2, 1(nipApiii , ),(21npppP P), 2, 1(nipApiii ),(),(21

3、21nnApApAppppAAP ),(2211nnppp , P ),(21nppp n 211 P,1 APP AnAn,n ,21 A n 21A, 22 , 41 ,11 APP.1 A 1513 A,11 1p.51 2p,2004 1 ,5111 ),(21ppP,21 APPA2 ,4002 2 ,1511 ),(12ppP, 21 . 132 321,pppA 163053064 A,111 1P,012 2P.100 3P 101011021 ),(321pppPA 121011021 1P.100010002 APP1An,A,P APP1Ai inA A), 2, 1()

4、(ninnEArii nAA, 0, 0, 1, 2 in, 0 ,2)()0(innBrEBr B B 000100011, 0, 0, 1, 2 in, 0 ,1)()0(innArEAr A 000000111A;,21s A,i in0)( xAEi iniinii ,21iinii ,21i ,111211n ,21ssnss ,222221n An) ,(21222211121121ssnssnnP APP1 .2211 ss 0 , 221 . 73 221 , 0)(1 xEA 242422221 A |EA 242422221)7()2(2 04420442022321321

5、321xxxxxxxxx,102 1p.110 2p, 73 0)(3 xEA 321,pppA211210102 , 0 TP)2, 2 , 1(3 AaA, 11 , 11 A, 132 . 132 ,00111100 a A 011110 a |AE )1()1(2 0)( xAEAE , 1)( AEr, 1)( AEr. 1 a, 01 a1 aA 10101101a AE,000100101 a1. 1. 利用矩阵对角化计算矩阵多项式利用矩阵对角化计算矩阵多项式A,BnAB P1 PBPA,1 PPBAkkAEaAaAaAnnn 10)( 11110 PEPaPBPaPPBannn

6、110)( PEaBaBaPnnn,)(1 PBP APP1P,1 PPAkk ,)()(1 PPA k ,21 knkk )( ,)()()(21 n A. )(A .0)( f)( fAA P A, ),(211ndiagAPP i A.0)( if ,1APPA .1OPOP PPPfAf 1)()( 1 P )()(1nff )(ixi )0(x,5 .135 . 1 A 99. 010. 001. 090. 090. 011 a10. 021 a 515.13485. 1 5 .135 . 199. 010. 001. 090. 0 )0()1(Axx)0()1()()0(2)1()2(,xAAxxxAAxxnnn ,)0(10)10(xAx 10AA0890. 089. 12 AE 99. 01 . 001. 09 . 0 , 11 .89. 02 ,21 A11 ,0 xAE .101 1P89. 02 代入代入 ,0 xAE 得其对应特征向量为得其对应特征向量为 2P.11 有有 21ppP令令,11011 APP1,89. 0001 ,1 PPA ,11010 PPA 而而 11011111 110111111 P)0(110)10(xPPx 111 5 .135 . 11101189. 000111011