I want to prove that if $A \otimes B = (C \otimes D)Z$ where $A,C \in M_m$ and $B, D \in M_n$ and $Z \in M_{m \times n}$ is invertible then there are invertible matrices $X$ and $Y$ such that $A = CX$ and $B = DY$.
I am stuck with trying to prove that $Z$ has to be equal to $X' \otimes Y'$ for some $X' \in M_m ,Y' \in M_n$ (I am pretty sure this is necessarily true). I have got as far as partitioning $Z$ into $m$ number of $n \times n$ matrices $Y_{ij}$, and have shown (hopefully correctly) that all the $DY_{ij}$ are scalar multiples of one another, I want to show that the $Y_{ij}$'s are scalar multiples of one another.
Any hints would be much appreciated. Apart from showing that $Z$ has to be equal to $X' \otimes Y'$ for some $X',Y'$, I am okay with the rest of the problem.