Given a positive definite matrix $M\in\mathbb{R}^{(2n)\times(2n)}$ of the following block form
$$M = \begin{bmatrix}A & X\\X^\top& B\end{bmatrix},$$
where $A, B\in\mathbb{R}^n$ are positive definite.
My question is: is it possible to find the relation between $XX^\top$ or $X^\top X$ and $A, B$. Something like $XX^\top+X^\top X\leq A^2+B^2$.
Any help would be highly appreciated!