# Eigenvalues and eigenvectors of interleaved “circulant like” matrices

Let $k|n$ and $k\geq 2$ and consider $k$ 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

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

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

2. What can be said about, say, the eigenvalues/eigenvectors of the matrix $$[A|B]_2^T \cdot [A|B]_2$$ which I think is a symmetric matrix?

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.