Say I have matrix product as follows:
$X.A$
where $X$ and $A$ are $n\times n$ matrices and $A$ is invertible.
Can all such products be represented as:
$B.X$
where $B$ is some $n\times n$ matrix and a transformed version of $A$?
(If not all, what conditions are needed for being able to do so?)
NOTE: I don't need a specific transformation to get to B. Just a proof that such a rearrangement is possible is enough.
THINGS I HAVE TRIED: I had begun by assuming $X.A = B.X$, and vectorizing both sides which leads to the following equation:
$\big[(A^T\otimes I_n ) - (I_n \otimes B)\big] vec(X) = 0_{n^2}$
where $\otimes$ is the Kronecker product.
I haven't been able to go any further!
EDIT 1: Edited the question - $X$ is not necessarily invertible
EDIT 2: Sylvester Equation is somewhat similar to the problem I'm trying to solve.
You want $$XA=BX$$
Solve for $B$ and you get $$ B=XAX^{-1}$$
if your matrices are invertible then there is no problem finding $B$