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 $A$.