I am trying to find some properties about the eigenvalues of the following Kronecker sum
$$ M = (\mathbb{I}_{\nu} \otimes A) + (B \otimes \mathbb{I}_{\nu})$$
where $A, B\in\mathbb{R}^{n\times n}$, $\mathbb{I}_{\nu}$ denotes the $\nu \times \nu$ identity matrix and $\otimes$ denotes the Kronecker product.
I am 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 $\nu \neq n$.