Suppose we have a symmetric block matrix $A$ where the blocks $A_{ii}$ along the diagonal have known eigenvalues.
$$A= \begin{pmatrix}A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \cdots & A_{nn} \end{pmatrix}$$
Is there a way to find the eigenvalues of $A$?
Similar question 1: In the special case where the off-diagonal matrices are 0, then the eigenvalues are easily found.
Similar question 2: However, in the case where the diagonals are zero and there are no other constraints on the matrices, it seems we cannot find the eigenvalues. I am hoping my case is easier since all the matrices are symmetric with diagonals not necessarily zero.
