# Block matrix of order $m$ with three block matrices

How to find eigenvalues of following block matrices?

$M=\begin{bmatrix} A & B & O & O & O & O & O & \cdots & O & O\\ B & A & B & O & O & O & O & \cdots & O & O\\ O & B & A & B & O & O & O & \cdots & O & O\\ O & O & B & A & B & O & O & \cdots & O & O\\ O & O & O & B & A & B & O & \cdots & O & O\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \cdots & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots\\ O & O & O & O & O & O & O & O & A & B\\ O & O & O & O & O & O & O & O & B & A\\ \end{bmatrix}_m$

Where,

$A=\begin{bmatrix} 0 & 1 & 0 & 0 & 0 &\cdots & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & \cdots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & \ddots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots\\ 0 & 0 & 0 & 0 & 0 & \ddots & 0 & 1\\ 1 & 0 & 0 & 0 & 0 & \cdots & 1 & 0\\ \end{bmatrix}_n$

$B=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 &\cdots & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{bmatrix}_n$

$O=\begin{bmatrix} 0 & 0 & 0 & 0 & 0 &\cdots & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{bmatrix}_n$