Can a matrix M be written as A*B*C where A and C are symmetric positive definite?

Question

Given a matrix $M$ and $B$ (not necessarily square), is there a way to determine whether symmetric positive definite matrices $A$ and $C$ exist such that $M=A B C$ ?

Answer 1

I don't think I have an answer to the very general question that you've asked. Here's the answer to a simpler version which might be helpful.

If $G_p = [g_{ij}]$ is the $n\times n$ matrix with elements

$g_{ij} = \left \{ \begin{array}{ll} 1, & i=j\leq p \\ -1, & i=j > p \\ 0, & i \neq j \end{array} \right .$

then let $O(p,q)$ be the set (it turns out to be a group) of matrices $A$ such that

$A^T G_p A = G_p$.

It can be shown that if $S$ is a diagonal complex matrix such that

$I = S^T G_p S$

then every member, $A$, of $O(p,q)$ can be written in the form

$A = S B S^{-1}$

where $B$ is a complex orthogonal matrix.

So, this doesn't answer your question but it is certainly closely related. $O(p,q)$ is called the "pseudo-orthogonal group of type (p,q)", so looking that up may give you some leads towards what you are looking for.