I am trying to find the eigenvalues of a real circulant matrix, $C$, plus a real diagonal matrix, $D$. My approach has been to successively apply the matrix determinant lemma by viewing $$D = \sum_{i=1}^n d_i e_i e_i^T$$ where $d_i$ are the entries of $D$ and $e_i$ is the $i$-th canonical vector. Is something like this even possible? I've searched everywhere and nothing seems to pop up. If I'm able to find a closed form solution (or an efficient numerical procedure) I'd like to generalize it to the case where $C$ is block circulant and $D$ is block diagonal (same block sizes).
Any insights or references is appreciated.