Q: Let $m, n$ be positive intergers. Let $A$ and $B$ real $n\times n$ matrices. Assume that $B$ is symmetric and positive definite. If $A$ commutes with $B^{m}$, prove that $A$ commutes with $B$.
So I don't really know where to start with this other than the decomposition theorem for symmetric positive operators, so that I can write $B$ as diagonal with positive real entries. Perhaps taking the $m^{th}$ root could be useful which we can do since $B$ has positive entries, but I'm not sure.
Many thanks!