Let $R = \mathbb{Z}[x_{1}, \dots, x_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear combinations of entries in $X$. 
Let $m \ge n$ and $r \ge n^{2}$.

What is the best way to compute matrices $A$ and $B$ such that $AXB = Y$? Any linear algebra tools useful here?