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I have a matrix that is given by $A e^{i q} + A^* e^{-i q}$ with $A$ a nilpotent $n\times n$ matrix. 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)

I have a matrix that is given by $A e^{i q} + A^* e^{-i q}$ with $A$ a nilpotent $n\times n$ matrix. The eigenvalues I get turn out always to be independent of $q$ 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 $q$).

I have a matrix that is given by $A e^{i q} + B e^{-i q}$ with both $A$ and $B$ nilpotent $n\times n$ matrices. 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)

I have a matrix that is given by $A e^{i q} + A^* e^{-i q}$ with $A$ a nilpotent $n\times n$ matrix. 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)