Eigenvalues of Matrices Raised to Powers

Question

What is the relationship between eigenvalues of some square matrix $A$ and the eigenvalues of $A^k$ for some positive integer $k$? How about eigenvectors? I haven't touched linear algebra in a while and am pretty shaky on this stuff.

Answer 1

Something to consider: If $Av = \lambda v$, then $A^2 v = A(Av) = A(\lambda v) = \lambda (Av) = \lambda^2 v$, then $A^3 v = A^2 (A v) = A^2 (\lambda v) = \lambda (A^2 v) = \cdots$