%0 Journal Article
%T On the least singular value of random symmetric matrices
%A Hoi H. Nguyen
%J Mathematics
%D 2011
%I arXiv
%X 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 random symmetric matrix whose upper diagonal entries $x_{ij}$ 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 $P(\sigma_n(M_n)\le n^{-A})\le n^{-B}$. The proof uses an inverse-type result concerning concentration of quadratic forms, which is of interest of its own.
%U http://arxiv.org/abs/1102.1476v2