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Random symmetric matrices are almost surely non-singular

Kevin Costello·Terence Tao·Van Vu
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Abstract

Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability $1-O(n^{-1/8+δ})$ for any fixed $δ> 0$. The proof uses a quadratic version of Littlewood-Offord type results concerning the concentration functions of random variables and can be extended for more general models of random matrices.

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Kevin Costello, Terence Tao, Van Vu. 2005-05-09. Random symmetric matrices are almost surely non-singular. https://arxiv.org/abs/math/0505156

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