Similar Powers of a Matrix

Let $p,q$ be coprime integers such that $|p|+|q|>2$. We characterize the matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^p$ and $A^q$ are similar. If $A$ is invertible, we prove that $A$ is a polynomial in $A^p$ and $A^q$. To achieve this, we study the matrix equation $B^{-1}A^pB=A^q$. We show that for such matrices, $B^{-1}AB$ and $A$ commute. When $A$ is diagonalizable, $A$ is a root of $I_n$ and $B^{-1}AB$ is a power of $A$. We explicitly solve the previous equation when $A$ has $n$ distinct eigenvalues or when $A$ has a sole eigenvalue. In the second part, we completely solve the $2\times{2}$ case of the more general matrix equation $A^{r}B^{s}A^{r'}B^{s'}=\pm{I}_2$.