An upper bound on the smallest singular value of a square random matrix

Abstract

Let $A=(a_{ij})$ be a square $n\times n$ matrix with i.i.d. zero mean and unit variance entries. Rudelson and Vershynin showed that the upper bound for a smallest singular value $s_n(A)$ is of order $n^{-\frac{1}{2}}$ with probability close to one under additional assumption on entries of $A$ that $\mathbb{E}{a}_{11}^4<\infty$. We remove the assumption on the fourth moment and show the upper bound assuming only $\mathbb{E}{a}_{11}^2=1.$

14 pages; A few typos corrected