Let $A$ be an $m\times n$ matrix such that $m < n$. I would like to know the conditions on $A$ such that the following is true:
$$\|Ax\| \leq \|Ay\| \implies \|x\| \leq \|y\|$$
It can easily be shown that if $\kappa(A)=1$ (condition number) then this property is satisfied. I am looking for the most general type of matrices that satisfy this condition.
Any help is much appreciated.
Thanks, Phanindra