Let $ k|n$ and $ k\geq 2$ and consider $ k$ symmetric matrices $ $ A,B,\ldots,K \in \mathbb{R}^{n\times n}.$ $ Consider a $ k-$ interleaving of these matrices where we take every $ k^{th}$ row in order from the given matrices. I will illustrate with an example for $ k=2.$

Let $ $ A=\begin{bmatrix} a_0& a_1 & \dots & a_{n-1}\ a_{n-1} & a_0 & \dots & a_{n-2}\ \ddots & &\ddots &\ddots \ a_1 & a_2 & \dots & a_0 \end{bmatrix},$ $ and

$ $ B=\begin{bmatrix} b_0& b_1 & \dots & b_{n-1}\ b_{n-1} & b_0 & \dots & b_{n-2}\ \ddots & &\ddots &\ddots \ b_1 & b_2 & \dots & b_0 \end{bmatrix},$ $

then a $ 2-$ interleaving of these matrices which is still $ n\times n$ is

$ $ [A|B]_2=\begin{bmatrix} a_0& a_1 & \dots & b_{n-1}\ b_{n-1} & b_0& \dots & b_{n-2}\ a_{n-2} & a_{n-1} & \dots & a_{n-3}\ b_3 & b_4 & \dots & b_{2}\ \ddots & &\ddots &\ddots \ a_2 & a_3 & \dots & a_1\ b_1 & b_2 & \dots & b_0 \end{bmatrix}.$ $

What can be said about the eigenvalues/eigenvectors of the interleaved matrix in terms of those of the original matrices?

It is well known that the columns of the $ n\times n$ DFT matrix are the eigenvectors and the DFT coefficients of the first row are the eigenvalues for a circulant matrix.