Electronic Journal of Probability

On the least singular value of random symmetric matrices

Hoi Nguyen

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Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a {\it random symmetric matrix} whose upper diagonal entries $x_{ij}, 1\le i\le j,$ are iid copies of a random variable $\xi$. Under a very general assumption on $\xi$, we show that for any $B>0$ there exists $A>0$ such that $\mathbb{P}(\sigma_n(M_n)\le n^{-A})\le n^{-B}$.

Article information

Electron. J. Probab. Volume 17 (2012), paper no. 53, 19 pp.

Accepted: 15 July 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 15A52
Secondary: 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx] 11B25: Arithmetic progressions [See also 11N13]

Random symmetric matrices least singular values

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Nguyen, Hoi. On the least singular value of random symmetric matrices. Electron. J. Probab. 17 (2012), paper no. 53, 19 pp. doi:10.1214/EJP.v17-2165. http://projecteuclid.org/euclid.ejp/1465062375.

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