Solve Eigenvalue Problem: q, x, A, Ak

Lanthanum
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Homework Statement


Given that q is an eigenvalue of a square matrix A with corresponding eigenvector x, show that qk is an eigenvalue of Ak 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 (Ak-qkI)x=0, and this is where I'm having trouble.
If anyone has time to help I would really appreciate it.
 
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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)
 

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