# Eigenvalues of a (non-traditional) Kronecker sum

I'm trying to find some properties about the eigenvalues of the following operation. Let $A \text{ and }B\in\mathbb{R}^{n\times n}$ and consider $$M = (\mathbb{I}_{\nu} \otimes A) + (B \otimes \mathbb{I}_{\nu})$$ where $\mathbb{I}_{\nu}$ denotes and identity matrix of dimension $\nu$ and the symbol $\otimes$ stands for the usual Kronecker product.

I'm interested in the location of the eigenvalues of $M$, since I need some conditions to guarantee the existence of the inverse. In this case, the well-known result about the Kronecker sum would only apply if $\nu = n$.

In fact, in my case, this last statement does not hold, and I'm interested in a more general situation where $\nu \neq n$.

• Probably you meant $B\in\mathbb{R}^{n\times n}$, and not $W$! – Davide Morgante Aug 4 '18 at 12:47