Random matrices: Probability of Normality

TL;DR

This paper studies the probability that a random matrix with i.i.d. Rademacher entries is normal, providing bounds that decay exponentially with the size of the matrix.

Contribution

It establishes bounds on the probability of normality for random Rademacher matrices, a novel probabilistic analysis in random matrix theory.

Findings

Probability of normality decreases exponentially with matrix size.

Provides explicit bounds for the probability.

Conjectures the lower bound is tight.

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 $\{ \pm1 \}$ 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.