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.