Solve Eigenvalue Problem: q, x, A, Ak

[Lanthanum]

Homework Statement

Given that q is an eigenvalue of a square matrix A with corresponding eigenvector x, show that q^k is an eigenvalue of A^k and x is a corresponding eigenvector.

Homework Equations

N/A

The Attempt at a Solution

I really haven't been able to get far, but;

If x is an eigenvector of A corresponding to q, then;
0=(A-qI)x
To complete the proof I need to use this equation to show that (A^k-q^kI)x=0, and this is where I'm having trouble.
If anyone has time to help I would really appreciate it.

 

[hgfalling]

I think you might want to use a different definition of an eigenvalue. Then the proof is really easy, with like hardly any algebra at all.
(edit: well, rearrange the one you have, I guess)