On the singularity of random matrices with independent entries

TL;DR

This paper proves that the probability of an n by n matrix with independent, non-degenerate entries being singular decreases at a rate of O(1/√n), using a simple Bernoulli decomposition approach.

Contribution

It provides an elementary and concise proof of the singularity probability bound for matrices with independent, non-identically distributed entries.

Findings

Probability of singularity is O(1/√n)

Elementary proof using Bernoulli decomposition

Applicable to non-identically distributed entries

Abstract

We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose of this note is to provide a short and elementary proof of this fact using a Bernoulli decomposition of arbitrary non degenerate random variables.