On the least singular value of random symmetric matrices

Hoi H Nguyen (University of Pennsylvania)


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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Pages: 1-19

Publication Date: July 15, 2012

DOI: 10.1214/EJP.v17-2165


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