How to prove the Eigenvalue property

[claudiocamera]

Hi there.

I was wondering how to prove the proposition below:

If S is nonsingular and B=SAS-1, then the eigenvalues of A and B coincides.

Any help ?

 

[coros]

Re: Eigenvalue property

From the eigenvalue property:

Bx = λx, then we substitute B with SAS-1

SAS-1x = λx, we multiply both sides by S-1 on the left ( S is nonsingular so S-1 exist ) and we get

AS-1x = λS-1x, because λ is a number, so A has the same eigenvalue λ and the eigenvector is S-1x