# Rewrite kronecker product of identity plus something

I'm working on trying to find a way to get the eigen-values of a complicated matrix but all the original elements themselves are either block-diagonal (as in, all blocks are the same also) or some simple repeated matrix. Computationally, I see a pattern of repeated eigen-values that makes me think I could write them in a more analytical form. I'm however getting into issues because there's a $$I_n - D$$ like term that's making it troublesome for me. This question is a smaller piece from this though.

Just to be clear on some notation. Let $$J_k = 1_k1_k'$$ be the square matrix of size $$k$$ of all 1s. Let $$A$$ and $$B$$ be square matrices of size $$c$$; $$A,B\in\mathbb{R}^{c \times c}$$.

Is there anyway to rewrite the following sum of kronecker products in such a way that find eigen-values would be "simple"?

$$\left( I_k \otimes A \right) + \left(J_k \otimes B \right)$$

where all matrix multiplication is possible. I think if it's possible it has something to do with a clever way of making $$I_k$$ and $$J_k$$ look more similar to each other but I haven't been able to think of anything.

Specifically in my case $$A$$ and $$B$$ have some similar structure in that $$A=Z'WZ$$ and $$B=Z'WCZ$$ but any points on the more general problem may be helpful.

There is a "smaller" piece earlier of the form $$I_{kc} - J_k \otimes D$$, $$D\in\mathbb{R}^{c\times c}$$, that if I had some other way to rewrite might make the later things simpler.

Edit:

I guess consequently that I'm asking about how to "simplify" the eigen-values of a block-matrix of the form

$$\begin{pmatrix} A & A + B \\ A + B & A \end{pmatrix}$$

which I don't think exists. Perhaps I've answered myself that the solution is "no" but maybe there's something else I'm missing.