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As for the eigenvalues there is neat trick I learned once: A matrix of the form $\left(\begin{smallmatrix} A & B \\ B & A\end{smallmatrix}\right)$ is conjugate (by $\left(\begin{smallmatrix} I & I \\ I & -I \end{smallmatrix}\right)$) to $\left(\begin{smallmatrix} A+B & \\ & A-B\end{smallmatrix}\right)$. Therefore there is a recursion for the (multi)set of eigenvalues of your matrix.