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I have the following matrix equation:

$AXB+CXD=E$

with A B C D E constant matrices and X an unknown matrix.

How can I isolate X?? I'm pretty sure that is basic but I don't know how to do it. Can anyone show me the basic rules for rearranging everything, if any?

Thanks!

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    $\begingroup$ Related: Sylvester equation. Note that your equation can be put in this form as $(C^{-1}A)X+X(DB^{-1})=C^{-1}EB^{-1}$ as long as $B,C$ are invertible. $\endgroup$
    – user856
    Commented Oct 10, 2019 at 12:52

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It's not so simple.

You can consider $X \to A X B + C X D$ as a linear operator on matrices of whatever size $X$ is. Thus if your matrices are all $n \times n$, rearrange them into $1 \times n^2$ vectors and your equation becomes an $n^2 \times n^2$ linear system.

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