Question

I have a matrix that is given by $A e^{i q} + B A^* e^{-i q}$ with both $A$ and $B$ a nilpotent $n\times n$ matricesmatrix. The eigenvalues I get turn out always to be independent on q but I cannot prove it. I want to know how can I prove this (or if this alone is not sufficient for the eigenvalues to be independent on q)