Suppose $A$ and $B$ are two $n \times n$ real matrices, with the special property that both $A$ and $B$ only have one eigenvalue, say equal to a real number $\lambda \ne 0$. That is, if $Av = \mu v$ for any complex number $\mu$, then actually $\mu = \lambda$, and similarly for $B$: if $Bw = \sigma w$ for some complex number $\sigma$ then actually $\sigma = \lambda$.
What can you say about the eigenvalues of $A^i B^j$ for $i,j \in {1,2,\dots}$? In general, the answer is nothing I think, what extra conditions might one need to put on $A$ and $B$ in order to say something?