Random symmetric matrices are almost surely non-singular

Details Random symmetric matrices are almost surely non-singular Random symmetric matrices are almost surely non-singular

2005-05-09

arxiv.org/abs/math/0505156v1

Mathematics

Probability

16 pages, no figures, submitted, Duke Math J

Scientific paper

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 $1-O(n^{-1/8+\delta})$ for any fixed $\delta > 0$. 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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