A square matrix $M$ is called a $P$-matrix ($P$ stands for "positive") if all the principal minors of $M$ are positive. My question is the following.
Let $\mathscr{M}$ be a class of $n\times n$ real matrices such that $\forall I \subset \{1,\cdots,n\}, \forall M\in\mathscr{M}$, the principal minor of $M$ corresponding to $I$ has the prescribed sign (depending on $I$). Under what assumption can one guarantee there exist real matrices $A$ and $B$ such that, for all $M\in \mathscr{M}$, the matrix $AMB$ is a $P$-matrix?
The assumptions I was expecting are upper bound on the entries and lower bound on the principal minors. But there is no reason to exclude other possibilities.
Thanks!