We know how to factorize a positive definite matrix, say $M$, by $A$ such that $AA^T=M$.
Say that we have three matrices $M_i$, $i=1,2,3$, and I want to have two matrices $AA^T = M_1, BB^T = M_2$, such that $$ AB^T = M_3 $$
The $M_3$ being some kind of covariant constraint for the two matrices.
Is there any work to study such a problem, for example, when does a solution exist? How to solve it?
Thanks!