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arXiv · 1711.02842

Random matrices: Probability of Normality

Andrei Deneanu·Van Vu
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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.

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Andrei Deneanu, Van Vu. 2017-11-08. Random matrices: Probability of Normality. https://arxiv.org/abs/1711.02842

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