Exercise 7, § 5

Let $A$ be a matrix with $m$ rows and $n$ columns, of rank $r$, over a field $K$. If $P$ is a square matrix of order $m$ and $Q$ is a square matrix of order $n$ over $K$, such that $PAQ = A$, show that there exists a matrix $P'$ similar to $P$ and a matrix $Q'$ similar to $Q$ such that

where $R$ is an invertible square matrix of order $r$, $U$ (resp. $U'$) is a square matrix of order $m - r$ (resp. $n - r$), and $S$ (resp. $S'$) is a matrix with $r$ rows and $m - r$ columns (resp. $n - r$ rows and $m$ columns).