Eigenvalues of a block matrix from the eigenvalues of the blocks

Question

I am trying to find the eigenvalues of the following complex matrix \begin{align} M=\left(\begin{matrix} A & B \\ B^\dagger & A^\dagger \end{matrix}\right) \end{align} where the symbol $\dagger$ stands for the conjugate transpose of the matrix.

Is there an expression of the eigenvalues of $M$ as a function of $A$ and $B$, or their properties?

Or an upper bound?

Feel free to make some assumptions on $A$ and $B$, this paper for instance, requires positive definiteness of the diagonal blocks. I haven't been able to find further research yet.

Answer 1

If $A=\pmatrix{0&0\\0&0}$ and $B=\pmatrix{0&x\\0&0}$, for a real number $x$, then all the eigenvalues of $A$ and $B$ are zero, but the eigenvalues of $M$ are $x,-x,0,0$.

So the eigenvalues of $M$ are not determined by, or bounded by, the eigenvalues of $A$ and $B$.