Given a $n\times m$ rectangular matrix A over a field $k$, does there exist a $m\times q $ matrix $Q$ such that $QQ^T=I$ and $AQ$ has only integer elements.
This does not hold in general. A simple counterexample is the matrix \begin{equation} A=\begin{bmatrix}1&0\\0&1/2\end{bmatrix} \end{equation}
If we consider a $2\times n$ matrix $Q$ \begin{equation} Q=\begin{bmatrix}a_1&a_2&... & a_n\\b_1&b_2 & ... & b_n\end{bmatrix} \end{equation} then we have \begin{equation} AQ=\begin{bmatrix}a_1&a_2&... & a_n\\b_1/2&b_2/2 & ... & b_n/2\end{bmatrix} \end{equation} For $AQ$ to have integer elements we must have that $b_i\geq2$, but this is not compatible with the constraint that $QQ^T=I$.
So my question is when is such a representation possible? Are there any meaningful criteria?