The least singular value of a random square matrix is O(n^{-1/2})

TL;DR

This paper proves that the smallest singular value of a random square matrix with i.i.d. entries is of order n^{-1/2} with high probability, matching previous lower bounds and confirming the typical scale.

Contribution

The authors establish a matching upper bound for the smallest singular value of such matrices, completing the understanding of its typical magnitude.

Findings

Smallest singular value is of order n^{-1/2} with high probability

Matching upper and lower bounds confirm the typical scale

Results apply to matrices with i.i.d. centered entries with unit variance

Abstract

Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate.