Random symmetric matrices are almost surely nonsingular

Random symmetric matrices are almost surely nonsingular

Let Q n denote a random symmetric ( n × n ) -matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1 / 2 ). We prove that Q n is nonsingular with probability 1 - O ( n - 1 / 8 + δ ) for any fix...

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Veröffentlicht in:Duke mathematical journal 2006-11, Vol.135 (2), p.395-413
Hauptverfasser: Costello, Kevin P., Tao, Terence, Vu, Van
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05D40
15A52
Probabilistic methods
Random matrices
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Tao, Terence
Vu, Van
description Let Q n denote a random symmetric ( n × n ) -matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1 / 2 ). We prove that Q n is nonsingular 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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15A52
Probabilistic methods
Random matrices
title Random symmetric matrices are almost surely nonsingular
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