Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists matrices $A$, $B$ such that
That is, for any $N\in S_2$, there is some $M\in S_1$ such that $AMB=N$.
Since $S_1$ and $S_2$ are symmetric matrix spaces, could we have some restriction on $A$ and $B$.
The question is that could we find some $C$ such that