Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in the literature that talks about properties of this kind of matrices? 


I have seen that there are some results for other kind of random matrices (for example matrices whose entries are i.i.d. Gaussian.) but not for this simple matrix of $\pm 1$.


I would be interested for example on the distribution of the $\sigma\_{\max}(A)$, but not in an asymptotic regime, as $m$, $n$ are finite numbers and usually small in my case.


Thank you very much for any pointer or any thoughts.



Maximum singular value of a random $\pm 1$ matrix
<http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf>
Theorem 5.39 (page 23) gives a non-asymptotic upper bound on the largest singular value


