Rewriting the matrix equation $AX = YB$ as $Y = CX$? Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have to estimate the two independent polarizations of a signal given the data from multiple detectors and the scalar response function for each polarization (LIGO data analysis). Being able to rewrite the equation above in general would make it possible to express fitted values as seen from multiple detectors (Y) as an explicit function of X, the data seen at each detector, and would enable a generalized cross-validation calculation to choose a regularization parameter.
Thanks.

In general it is not going to be possible.
Example: Let
$
A = Y =
\begin{pmatrix}
1 & 1 \\ 
0 & 0
\end{pmatrix}
$
and
$
X = B =
\begin{pmatrix}
1 & 0 \\ 
1 & 0
\end{pmatrix}.
$
Then $AX=YB$, but no matter what $C$ you choose, $CX$ has $0$'s in its second column and thus cannot equal $Y$.
