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 $\nu \neq n$.

Thanks in advance!


  • $\begingroup$ Probably you meant $B\in\mathbb{R}^{n\times n}$, and not $W$! $\endgroup$ – Davide Morgante Aug 4 '18 at 12:47
  • $\begingroup$ @DavideMorgante thanks a million! I'll change it right away. $\endgroup$ – Nico F. Aug 4 '18 at 13:12

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