If $A$ has an eigenvector $x$ with eigenvalue $\lambda$, find an eigenvector for the matrix $B = S^{-1}AS$. Find the corresponding eigenvalue as well.
Do: $(B-\lambda I)x = 0$
($S^{-1}AS - \lambda I)x = 0$. I am kind of stuck here...
I was thinking about an expression for $A^{-1}$ and minimal polynomials. Is that a good place to start?