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

TL;DR

This paper establishes an upper bound on the smallest singular value of square random matrices with i.i.d. entries, removing previous moment assumptions and showing the bound holds under only finite second moment conditions.

Contribution

It proves an upper bound on the smallest singular value of square i.i.d. random matrices assuming only finite second moments, extending prior results that required finite fourth moments.

Findings

Upper bound of order n^{-1/2} for the smallest singular value.

Removal of the finite fourth moment assumption.

Bound holds under only finite second moment condition.

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 $n^{-\frac12}$ with probability close to one under additional assumption on entries of $A$ that $\mathbb{E}a^4_{11} < \infty$. We remove the assumption on the fourth moment and show the upper bound assuming only $\mathbb{E}a^2_{11} = 1.$