# On the linear transformation between matrix space

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

$$S_2\subset AS_1B.$$

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

$$S_2\subset CS_1C^T.$$

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I don't think your first claim is true as stated. If $n<m$ and $S_2$ contains a full-rank matrix, for instance, it is clear that one can't find $A$, $B$. Or if $S_1=\{0\}$ and $S_2\neq \{0\}$. Or maybe I am not understanding what you mean by "matrix space"? –  Federico Poloni Jun 15 '13 at 11:10
@Federico. I guess that this is not a claim, but a hypothesis. –  Denis Serre Jun 15 '13 at 12:28
@Denis Yes, thank you Denis, you are correct. –  gondolf Jun 15 '13 at 13:35
Thanks, now I get it after the edit. –  Federico Poloni Jun 15 '13 at 13:50