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.



"Additive version" of Kronecker product
**Federico** already mentioned the keyword. The precise answer may be found among others as [Theorem 13.16, of this book](http://www.siam.org/books/textbooks/OT91sample.pdf). (That theorem makes a restriction to real matrices, but that is not necessary).


