Diagonalization matrix

2010-12-16 10:42 pm

Find two distinct diagonal matrices D1 and D2 , and corresponding invertible matrices P1 and P2 such that P1−1AP1=D1 and P2−1AP2=D2 .
Put the smaller eigenvalue of A in the top left-hand corner of D1
I know how to find D2 and P2, but not D1 and P1, please help
P2=
1,1
-2,-1
D2=
-4,0
0,-8
thx

更新1:

o sorry, i forgot A is -12,-4 8,0

更新2:

hey lotus P1=[1 1//-1 -2], D1=[-8 0 //0 -4] check: P1^(-1)=[ 2 1 // -1 -1] P1^(-1)*A=[-16 -8// 4 4] P1^(-1)*A*P1=[-8 0 //0 -4] I tried P1=[1 1//-1 -2], D1=[-8 0 //0 -4] but I got P1^(-1)*A*P1= -4,0 -12,-8, this is not a diagonal matrix though :\

更新3:

Yea I got it I got it wrong because I put it in this way, thx P1*A*P1^(-1)

回答 (2)

Lotus_本願山彌陀講堂

2010-12-17 7:44 am

✔ 最佳答案

P2=[ 1 1// -2 -1], D2=[-4 0//0 -8]
A=P2*D2*P2^(-1)=[-12 -4 // 8 0 ]

P1=[1 1//-1 -2], D1=[-8 0 //0 -4]
check: P1^(-1)=[ 2 1 // -1 -1]
P1^(-1)*A=[-16 -8// 4 4]
P1^(-1)*A*P1=[-8 0 //0 -4]

2010-12-17 00:43:02 補充：
P1^(-1)*A*P1
=
[ 2 1 ] [-12 -4 ] [ 1 1 ]
[ -1 -1] [ 8 0 ] [ -1 -2]
=
[- 16 -8 ] [ 1 1 ]
[ 4 4 ] [-1 -2 ]
=
[ -8 0 ]
[ 0 -4 ]

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用戶9112

2010-12-17 5:34 am

What is A?

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