# Diagonalizing a symmetric block matrix

Let us consider the matrix

$$A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$ then this matrix has eigenvalues $$a\pm \sqrt{c^2+b^2}.$$

Now, let us consider a block matrix

$$A = \begin{pmatrix} A_1 & A_2 \\ A_2& A_1 \end{pmatrix},$$

then this block matrix can be block-diagonalized to

$$\begin{pmatrix} A_1-A_2 & 0 \\ 0& A_1+A_2 \end{pmatrix},$$ by conjugating it with $$1/\sqrt{2}\begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}.$$

I would like to know: Is there any similar construction that allows me to explicitly block-diagonalize

$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^*& A_1 \end{pmatrix},$$ where $$A_1$$ is a real-symmetric matrix?

This is a block-matrix version of the first scalar matrix that I have written down, but unlike that one, I do not see right away how to block-diagonalize $$A$$.

Say that the blocks are $$n\times n$$ (hence $$A$$ is $$2n\times2n$$). Your question amounts to finding explicit $$n$$-dimensional subspaces that are stable under $$M$$. Generically, such spaces are of the form $$E_M=\{(x,Mx);x\in{\mathbb C}^n\},$$ for suitable matrices $$M$$.
An elementary calculation shows that these matrices are the solutions of a Ricatti equation $$MA_2M+MA_1-A_1M-A_2^*=0_n.$$ This cannot be solved explicitly in general, but there are existence results and approximation procedures.
Mind that if $$M$$ is a solution, then so is $$N=-M^{-*}$$, and the spaces $$E_M,E_N$$ are transversal to each other, because $$M^*M$$ is positive semi-definite.