Suppose $A\in\mathbb{C}^{m\times m}$ has a singular value decomposition: $A=U \Sigma V^H$. Find diagonalization of a block hermitian matrix $B\in \mathbb{C}^{2m\times 2m}$, $B=[0 \; \; A^H; A \; \; 0]$.
I am kind of lost on this problem. I was thinking about writing $B=[0 \; \; (U\Sigma V^H)^H; U\Sigma V^H\; \; 0]$, but not sure where to go from there, I hope someone can help me get on the right track.
I've been searching on a similar question and found this approach helpful:
(Note that in the link, the $A^H$ and $A$ block matrices are swapped.)
I'm still a student myself, but since there are no other answers here yet, perhaps this will help you as well.
Good luck!