Let $F$ be an algebraically closed field, $n,s\ge1$ integers, and $B\in M_{ns}(F)$ an $ns\times ns$ matrix over $F$. Let $A_1, \ldots, A_n\in M_s(F)$ be $s\times s$ matrices over $F$, and let us consider the following polynomial $P(X)\in M_{ns}(F)[X]$: $$ P(X)= \begin{bmatrix} A_1 & \\ & A_1 & & \\ & &... & \\ & & & A_1 \end{bmatrix} + \begin{bmatrix} A_2 & \\ & A_2 & & \\ & &... & \\ & & & A_2 \end{bmatrix} X + \ldots + \begin{bmatrix} A_n & \\ & A_n & & \\ & &... & \\ & & & A_n \end{bmatrix} X^{n-1} $$ Given the eigenvalues of $B$, can we describe the eigenvalues of $B'=P(B)$?
If $s=1$, then $P$ can be viewed as a polynomial with scalar coefficients. This case was considered in this question, where it was shown that the eigenvalues of $B'$ are than obtained by applying the polynomial to the eigenvalues of $B$. I am interested in finding out whether this can be generalised to block matrices.
