Random matrices: Probability of Normality

Abstract

In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\{\pm 1\}$ with probability $1/2$) and prove $$2^{-\left(0.5+o(1)\right)n^2}\le P\left(M_n\text{ is normal}\right)\le 2^{-(0.302+o(1))n^2}.$$ We conjecture that the lower bound is sharp.