A matrix algebra question I have the following situation,
$$E = A X B + C X D$$
where $A,B,C,D,E$ and $X$ are matrices with proper dimensions. I want to obtain an expression like,
$$X = f(A,B,C,D,E)$$
i.e., leave $X$ alone at one side. Is there a way to do this? I am stuck. Thanks in advance!
PS: OK, my situation is more interesting. I have the following situation actually,
$$E = A^T X B^T + A X B$$
and try to solve above problem. Thanks!
 A: This is a problem of higher dimension than it would at first appear. You may use properties of Kroneker product to transform the equation into an equivalent higher dimension formula involving only left (or right) multiplications, thus giving a solution for $X$. This is how you would solve $E = XB + CX$ at least, have not tried it with the extra conditions you have.
Search Kroneker and/or vectorization. Your equation, if the proper inverses exist, is
$$A^{-1}ED^{-1} = XBD^{-1} + A^{-1}CX$$
This is indeed solved with the Kroneker product.
A: User adamW has already mentioned this. Here $X$ is your variable, say it is of size $N \times N$. So you have $N^2$ variables. Also observe the equations given are linear in terms of the entries of $X$. Now to bring out all of this, you need to use different properities of kronecker product and $vec$ operator. 
Your system can be expressed as a linear system of equations. The details are 
$vec(AXB)=(B^T\otimes A)vec(X)$
$vec(A^TXB^T)=(B\otimes A^T)vec(X)$
$vec(E)=(B^T\otimes A+B\otimes A^T)vec(X)$
Thus $vec(X)=(B^T\otimes A+B\otimes A^T)^{-1}vec(E)$
