I would like to show that every positive definite matrix $S \in \mathbb R^{2d \times 2d}$ can be written as: $$ S = M M^T, $$ where $M$ has the following form: $$ M = \Lambda \begin{bmatrix} I + AB & A \\ B & I \end{bmatrix}, $$ with some positive diagonal matrix $\Lambda \in \mathbb R_+^{2d \times 2d}$, and arbitrary matrices $A, B \in \mathbb R^{d \times d}$.
Optional: How do $A, B, \Lambda$ have to be picked given $S$?
Notes:
- From numerical experiments, I am pretty sure that it's true.
- I already know that the converse holds: For any given $A, B, \Lambda$ as above, $M$ has full rank, so $M M^T$ is positive definite.