Let $A$ and $B$ be $n\times n$ matrices over $\mathbb{C}$. If $AB=BA$, we know that we can simultaneously diagonalize $A$ and $B$ (or make them Jordan canonical form). What if they are weakly commutative in the sense that $AB=cBA$ for some $c\in \mathbb{C}^{\times}$? What can we say about $A$ and $B$?
I am sorry being a bit vague, but I came across with this kind of matrices in my little project and wonder what we can say about them.