I would like to find the inverse of the sum of a Kronecker product and a diagonal matrix. I found this answered here, but I don't see how the last step is valid.
Because of the fact that $C$ is diagonal, the spectral decomposition of $D^{−1}C$ itself is easy to obtain from that of $D^{−1}$.
It says that if we know the spectral decomposition of a symmetric matrix $D^{-1}$, we can easily find the same for $D^{−1}C$, where $C$ is diagonal. Can someone explain how to do this?
Since $D^{−1}C$ is not symmetric and not guaranteed to be orthogonally diagonalizable anymore, I don't quite see how this solution would work.
Thanks!