Eigenvectors and eigenvalues of $A$ and $B$ versus $A\cdot B$

Given two square matrices $A$ and $B$ of the same dimensions with entries in $\mathbb{C}$, as well as the sets of all eigenvalues and eigenvectors for both matrices $\{e_A,v_A\}$ and $\{e_B,v_B\}$, is there anything non-trivial we can conclude about the set of eigenvalues and eigenvectors $\{e_{AB},v_{AB}\}$ for the multiplied matrix $A\cdot B$?

• Bhatia's Matrix Analysis lists some useful inequalities to bound both eigenvalues and eigenvector variations. Most of them apply to $A \pm B$, but there are results for $AB$ as well. – Omnomnomnom Dec 20 '17 at 20:23

I don't know how trivial this is to you, but here it is:$$0\in e_{AB}\iff 0\in e_A\vee0\in e_B.$$
Here's another one: if $v\in v_A\cap v_B$, then $v\in v_{AB}$.
• I am more curious about relations such as writing $v_{AB}$ as a linear combination of $v_A$ or $v_B$ with perhaps a well defined procedure to fix the coefficients... But maybe that is too optimistic. – Kagaratsch Dec 20 '17 at 19:29