# Small ball probability for the condition number of random matrices

## TL;DR

This paper establishes a probabilistic bound on the ratio of the largest to smallest singular values of random matrices with i.i.d. subgaussian entries, providing new insights into their condition number behavior.

## Contribution

It presents a novel small ball probability estimate for the condition number of such matrices, combining existing techniques in a new way.

## Key findings

- Condition number bound: ${f P}igrac{s_{ ext{max}}(A)}{s_{ ext{min}}(A)} ext{ is small}ig] ext{ decays exponentially with } t^2.
- Singular value estimates: Probabilities for small singular values are tightly controlled, extending previous results.
- The estimate is new in the literature, despite being derivable from known results.

## Abstract

Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.