线性代数实对称矩阵的对角化

1、一、正交矩阵与正交变换二、施密特正交化方法三、实对称矩阵的性质四、实对称矩阵对角化的方法nA, ,EAAT A, ,1 AAT)1(; ;nTTEAAAA )2(A)(1 AAT或或)3()4()1()2(, ,TTAA)( 1 AAT)()(TTTABABABAB TTABBA)( TnAAE TAA , ,nE AB, ,nTEAA A)4(, , 1 AAT, ,AAAATT , ,AAT , , 12 A, , 1 A. .1 A)3(ABnAA, ,E )(21n , , , , TnTT 21, , , , ,)(21nA n , , , ,21AEAAT nTnTnTnnTTTn

2、TTT 212221212111 ijjTi EAAT EAAT , , , , jiji01. ., , , , , )21(nji PPxy Pxy yyyT PxPxTT xxT . .x 例例1 判别下列矩形是否为正交矩阵判别下列矩形是否为正交矩阵. .)1( 9/79/49/49/49/19/89/49/89/1,12/13/12/112/13/12/11 )2(r , , , ,21reee, , , ,21r , , , ,21r , , , ,21)1(, ,11 , , , ,1112122 , , , , , ,222321113133 , , , , , , , ,111

3、122221111 rrrrrrrrr ,21reee, , , ,)2(, ,111 e, ,222 e, ,rrre reee, , , ,21V;11 1211222|, 222231211333|,|, ,1 ,2 3 ,121 1 ,131 2 ,014 3 131 12164 ;111 35 014 12131 11135 .101 2 321,eee61|111 e,121 31|222 e,111 21|333 e.101 ),1 , 1 , 1 , 1(1 ),4 , 0 , 1, 1(2 ).1, 1 , 5 , 3(3 ,1 ,2 3 ),1 , 1 , 1 , 1(11

4、 1112122, )1 , 1 , 1 , 1(1111411)4 , 0 , 1, 1( ),3 , 1, 2, 0( )3 , 1, 2, 0(414)1 , 1 , 1 , 1(48)1, 1 , 5 , 3( ),0 , 2, 1 , 1( 3 22232111313, |111 e,21,21,21,21 )1 , 1 , 1 , 1(21 |222 e)3 , 1, 2, 0(141 ,143,141,142, 0 |333 e)0 , 2, 1 , 1(61 .0 ,62,61,61 A x,xAx , 0 x xxxAxA Ax x ,x )(AxxAxxTT xxT ,xx

5、T xAxAxxTT)( xxAT)( xxT)( ,xxT , 0)( xxT , 0 x niiiTxxxx1 niix120|, 0 , ,A0)( xEAi 0| EAi An21, ppA21, 111pAp ),0(1 p222pAp ),0(2 p)(21 ATTpp)(1111 TAp )(1 TTAp1 ,1ApT 21211)(pApppTT )(21AppT )(221ppT 212ppT 0)(2121 ppT 21 , 021 ppT1p2pAn ArEA )(EAr , rn rAn,P,1 APP Ansrrr,21).(21nrrrs i , 2 , 1( i)

6、 sA.,21s irirnrs nn,P 21rr PPAPP11, irsr ,1 ,s An. 1Ai );, 2 , 1(si . 20)( xEAi A. 3. 4. 5,P.1 APP,P0)5)(2)(1( , 11 A,320222021 A320222021 |AE 0 APP1 , 0)( xAE.)1 , 2 , 2(1Tp 11 , 22 . 53 , 0)5( xAE.)2 , 2, 1(3Tp .)2, 1, 2(2Tp , 0)2( xAE22 53 321,ppp321,ppp,3/ 13/ 23/ 2 |111pp ,3/ 23/ 13/ 2 |222pp 3

7、/ 23/ 23/ 1 |333pp 则则APPAPPT 1.500020001 令令,3/23/23/13/23/13/23/13/23/2 ),(321 P,P, 21 , 21 0)2( xAE,310130004 A2)4)(2( 310130004 |AE |. 432 ,110 1pAPP1 , 432 0)4( xAE,001 2p.110 3p321,ppp2p3p321,ppp)3 , 2 , 1(| ippiii ,2/12/10 1 ,001 2 .2/12/10 3 2/102/12/102/1010 ),(321 P.400040002 APP1A)2)(2)(2(

8、aa 0 a aa2020002 A |AE aa 2020002APP1 .P, 12 a; 3 a, 12 a. 1 a, 0 a. 3 a. 5 , 1 , 2AA, 21 , 0)2( xAE;)0 , 0 , 1(1Tp , 12 , 0)( xAE;)1, 1 , 0(2Tp , 53 , 0)5( xAE;)0 , 0 , 1(3Tp 321,ppp;)0 , 0 , 1(1T 3 ;21,21, 0T 2 ;21,21, 0T ,2/12/102/12/10001 ),(321 P.500010002 APP1.nAAA, .1 APP1 PPA.1 PPAnn),3)(1( A. 3, 121 ,2112 A |EA 2112342 P.3001 n n ,3001 ,