Let $n\_1>n\_2\geq 1$ be integers. Are there a known algorithms for generating $n\_2\times n\_1$-dimensional random matrices $A$ such that
$$
\|Ax - Ay\|<\|x-y\| \mbox{ if $x\neq y$}?
$$



Reference Request: Randomly Generated Contraction
[On some properties of contracting matrices:](https://arxiv.org/abs/math/0604457)  

For $n\_1=n\_2$ and if the norm is the $\|\cdots\|\_\infty$ norm, then the contractive property (with $\leq$ instead of $<$) is satisfied if the matrix is a [Markov matrix](https://en.wikipedia.org/wiki/Stochastic_matrix) (nonnegative real matrix elements with each row summing to 1) and moreover for every pair of rows there exists a column with nonzero entries in those two rows. With these restrictions a random matrix can be readily generated.


