1. ## matrix prove

Hi, all

If B = P^-1AP and Pv is an eigenvector of A then v is an eigenvector of B

I changed it to PB = AP and PBv = APv.

After that, what can I do?

I am sorry to bother you!

2. ## Re: matrix prove

Originally Posted by yanirose
After that, what can I do?
Use the fact that Pv is an eigenvector of A.

I wouldn't multiply B = P^(-1)AP by P.

3. ## Re: matrix prove

Originally Posted by yanirose
Hi, all

If B = P^-1AP and Pv is an eigenvector of A then v is an eigenvector of B

I changed it to PB = AP and PBv = APv.

After that, what can I do?

I am sorry to bother you!
$Pv$ is an eigenvector of $A$ so $A(Pv) = \lambda(Pv)$

$Bv = P^{-1} A Pv = P^{-1}\lambda (Pv) = \lambda P^{-1}Pv = \lambda v$

$Bv = \lambda v \Rightarrow v$ is an eigenvector of B.