I know that if $Av=\lambda v$, $A^nv=\lambda^nv$, so powers of matrices inherit eigenvectors. But suppose $A$ is a square $d$ by $d$ matrix with $k<d$ eigenvectors; could $A^n$ have more eigenvectors for some $n$? Suppose $A$ is invertible, does that change anything?
I'm really just trying to find out if, for $A\in GL_d(\mathbb{C})$, $A^n=I$ implies that $A$ has $d$ eigenvectors.