# Thread: similar matrices and eigenvalues

1. ## similar matrices and eigenvalues

Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.

I have already shown this using the characteristic polynomial but I have no idea how to do it this way.

Any help would be greatly appreciated!

2. Originally Posted by pseudonym
Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.

I have already shown this using the characteristic polynomial but I have no idea how to do it this way.

Any help would be greatly appreciated!

Supose $A\sim B\Longrightarrow A=P^{-1}AP\Longleftrightarrow PAP^{-1}=B$ for some invertible matrix $P$.
Now let $Av=\lambda v\,\,and\,\,Pv=w\Longrightarrow P^{-1}w=v$, so then:

$Bw=PAP^{-1}w=PAv=P(\lambda v)=\lambda Pv=\lambda w\Longrightarrow \lambda$ is an eigenvalue of B

Tonio