$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?

Question

$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?

Answer 1

You cannot do this in general unless you assume something stronger about the matrices. E.g. commutativity.

Naturally, if you are okay with $Y$ and its inverse appearing in the product, $AYBY^{-1}=X$.

Answer 2

Clearly, $AYBY^{-1}=XYY^{-1}=X$, but in general you cannot write $X$ only in terms of $A$ and $B$. For example, for $A=I$, $B=\left(\begin{matrix}1&0\\1&1\end{matrix}\right)$ and $Y=\left(\begin{matrix}1&1\\0&x\end{matrix}\right)$ (just picking some random values), $$X=AYBY^{-1}=\left(\begin{matrix}2&-x^{-1}\\x&0\end{matrix}\right)$$

which is clearly dependent on $x$.