# "Additive version" of Kronecker product

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.

• $A \otimes I + I \otimes B$ (sometimes called "Kronecker sum") should work. Commented Sep 28, 2015 at 14:03
• Or just $A\oplus B$ on the direct sum. Commented Sep 28, 2015 at 17:35
• @paul: that's not correct. You get the union (as a multiset) of the eigenvalues that way, not the pairwise sums. Commented Sep 28, 2015 at 21:24
• @QiaochuYuan, aha!, you are certainly right about that! I was not thinking! Commented Sep 28, 2015 at 23:39
• (I must note that I have seen sometimes the notation $A\oplus B$ used for the Kronecker sum as defined in my previous comment. This is, of course, very confusing, since it is also the standard notation for direct sums.) Commented Sep 29, 2015 at 4:11