Let $A$ and $B$ be two square matrices with complex entries. Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and $\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$. Then the Eigenvalues of the Kronecker product are exactly the products $\lambda_i \cdot \mu_j$. Is there an analogue for the sums of Eigenvalues? My precise question is the following:

For given natural numbers $m$ and $n$ are there polynomials $f_{rs} \in \mathbb{C}[x_{ij},y_{kl}: \, 1 \leq i,j \leq m, \, 1 \leq k,l \leq n]$ such that for every $n \times n$ matrix $A$ and every $m \times m$ matrix $B$ the Eigenvalues of the matrix $C=(f_{rs}(A,B))_{1 \leq r,s \leq mn}$ are exactly the sums of an Eigenvalue of $A$ and an Eigenvalue of $B$? Here $f_{rs}(A,B)$ stands for the complex number obtained by substituting $x_{ij}$ by the $(i,j)$th entry of $A$ and $y_{ij}$ by the $(i,j)$th entry of $B$.

I am aware of some similar construction where the matrix $C$ has the desired Eigenvalues *among others*. But for me it is important that they are no other Eigenvalues.