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2012 On the least singular value of random symmetric matrices
Hoi Nguyen
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Electron. J. Probab. 17: 1-19 (2012). DOI: 10.1214/EJP.v17-2165

Abstract

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 $\mathbb{P}(\sigma_n(M_n)\le n^{-A})\le n^{-B}$.

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Hoi Nguyen. "On the least singular value of random symmetric matrices." Electron. J. Probab. 17 1 - 19, 2012. https://doi.org/10.1214/EJP.v17-2165

Information

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

zbMATH: 1251.15014
MathSciNet: MR2955045
Digital Object Identifier: 10.1214/EJP.v17-2165

Subjects:
Primary: 15A52
Secondary: 11B25 , 15A63

Keywords: least singular values , random symmetric matrices

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