How are eigenvalues of block matrix related with whole matrix?

Question

For

$$A = \;\;\; \begin{pmatrix} A_{1,1} & A_{1,2} & A_{1,3} \\ A_{2,1} & A_{2,2} & A_{2,3} \\ A_{3,1} & A_{3,2} & A_{3,3} \\ \end{pmatrix} $$ where each of $A_{i,j}$ is a block matrix. What I wonder is the relation of eigenvalues of whole matrix $A$ and block matrix of $A_{i,j}$

If I know the each of eigenvalues of each block matrix $A_{i,j}$, are the eigenvalues of $A$ combined eigenvalues of each of $A_{i,j}$?

** All diagonal entries of $A$ is zero, and $A$ is symmetry matrix.

p.s) currently I am handling very big matrix $A$. It is too big to calculate eigenvalue by MATLAB. However, I can calculate the eigenvalues of block matrix $A_{i,j}$

---

Answer 1

No, they are not always related.

Hint : Take block matrices whose eigenvalues you can easily compute. Put it all into A above. Find out the eigen-values of A.

I tried with $2X2$ block matrices to obtain a $6X6$ matrix A. The eigen values for each block matrix were integers but A did not give me eigen-values which were all integers.