# Eigenvalue problem for symmetric block tridiagonal matrices?

Is there a procedure to find the eigenvalues of $\textbf{M}$? ‎ $$\begin{eqnarray} ‎\textbf{M}=\left[‎ ‎\begin {array}{ccccc}‎ ‎\textbf{A} & \textbf{B} & & &\\‎ ‎\textbf{B}^T & \textbf{ A} & \textbf{B} & &\\‎ ‎&\ddots &\ddots & \ddots &\\‎ ‎& & & & \textbf{B} \\‎ ‎& & & \textbf{B}^T & \textbf{A} ‎\end {array}‎ ‎\right]‎, ‎\end{eqnarray}‎‎$$

where $\textbf{B}^T$ is transpose of matrix $\textbf{B}$.

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Presumably, $A$ is symmetric. –  Denis Serre Oct 8 '12 at 20:28
How does this problem differ from the usual symmetric eigenvalue computation? Indeed, you should be able to calculate the eigenvalues faster than $n^3$ time because of the "tridiagonal" structure. –  Christopher A. Wong Oct 9 '12 at 22:16
Think about a 2 by 2 block matrix---seems to get its exact eigenvalues, knowing the eigenvalues of A and B does not really help that much (to get bounds, yes, but to get exact ones...) –  Suvrit Nov 6 '12 at 17:58