I have a $(2n)\times (2n)$ matrix defined in blocks:
$$ \begin{equation} \begin{split} M=\left[ \begin{array}{c|c} A & B\\ \hline C & D \\ \end{array} \right] \end{split} \end{equation} $$
where $A,B,C,D$ are all $n\times n$ matrices that are real, symmetric, and positive semi-definite.
Clearly $M$ would be diagonalisable if $B=C$. I am trying to understand if I can say anything about the possibility to diagonalise $M$ when $B \neq C$.
Any help or link to textbook would be highly appreciated.