If I have $N$ $M\times M$ symmetric positive definite matrices $A_i$ and an $N\times N$ positive semidefinite symmetric matrix B, let the $N\times N$ matrix $C_{ij}(\lambda)=B_{ij}$ for $i\ne j$ and $C_{ii}(\lambda)=B_{ii}+\lvert A_i\lambda 1_M\rvert$, so we are adding the characteristic polynomials of $A_i$ to the diagonal of $B$. Is there a simple construction – like a Kronecker product – that produces an $(NM)\times(NM)$ matrix with characteristic polynomial $P(\lambda)=\lvert C(\lambda)\rvert$?
