The smallest singular value of inhomogeneous square random matrices

Galyna V. Livshyts, Konstantin Tikhomirov, Roman Vershynin

Published 2019-09-10Version 1

We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq \frac{\varepsilon}{\sqrt{n}} \right) \leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0. $$ This extends earlier results \cite{RudVer-square, RebTikh} by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result \cite{Liv} where the matrix was required to have i.i.d. rows. Our model covers "inhomogeneus" matrices allowing different variances of the entries, as long as the sum of the second moments is of order $O(n^2)$. As a crucial element of the proof, we introduce the notion of the Randomized Least Common Denominator (RLCD) which allows to study anti-concentration properties of weighted sums of independent but not identically distributed variables.