I have a block tridiagonal Toeplitz matrix
$$M=\begin{bmatrix} A & Z & O\\ Y & A & Z\\ O & Y & A\end{bmatrix}$$
where
$$A=\begin{bmatrix} 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\end{bmatrix}, \qquad Y=\begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\end{bmatrix}, \qquad Z=\begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\end{bmatrix}.$$
I need to find its eigenvalues with out any scientific instruments. I know that the $k$-th eigenvalues of a tridiagonal Toeplitz matrix is
$$\lambda_k = A-2\sqrt{YZ} \cos\left(\frac{k\pi}{n+1}\right).$$
However, here, the elements in the matrix $M$ are blocks. How can I solve this?