<s>[INST] 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,
$A=\begin{bmatrix}
0 & 1 & 0 & 0 & 0  &\cdots & 0 & 1 \\ 
1 & 0 & 1 & 0 & 0 & \cdots & 0 & 0\\ 
0 & 1 & 0 & 1 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & 0 & 1 & \cdots & 0 & 0\\
0 & 0 & 0 & 1 & 0 & \ddots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots\\
0 & 0 & 0 & 0 & 0 & \ddots & 0 & 1\\
1 & 0 & 0 & 0 & 0 & \cdots & 1 & 0\\
\end{bmatrix}_n$
$B=\begin{bmatrix}
1 & 0 & 0 & 0 & 0  &\cdots & 0 & 0 \\ 
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
\end{bmatrix}_n$
$O=\begin{bmatrix}
0 & 0 & 0 & 0 & 0  &\cdots & 0 & 0 \\ 
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
\end{bmatrix}_n$
 [/INST] You can find a computation method for the eigenvalues and eigenvectors in the following recent document: "Eigendecomposition of Block Tridiagonal Matrices" by 
A. Sandryhaila and J.M.F. Moura (arxiv.org/pdf/1306.0217)
</s>