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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?

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 $r \ge n^{2}$.

Do there always exist $A$ and $B$ such that $AXB = Y$?

If so, what is the best way to compute matrices $A$ and $B$ such that $AXB = Y$? 

Any linear algebra tools useful here?

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 $m \ge n$.

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

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?