# Operator norm upper bound for sub-Gaussian tailed random matrices

## TL;DR

This paper establishes bounds on the operator norm of sub-Gaussian tailed random matrices, extending results to matrices with row-wise independence and non-uniform variances.

## Contribution

It proves that matrices with row-wise independent sub-Gaussian rows satisfy Tracy-Widom bounds, generalizing previous results that required independent coefficients.

## Key findings

- Operator norm bounded by O(√n) with high probability
- Row-wise independence suffices for Tracy-Widom bounds
- Results extend to non-uniform variance matrices

## Abstract

This paper investigates an upper bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients. However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance are not equal or for matrix for which coefficients are not independent. This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator norm remains bounded by $O(\sqrt n )$ with overwhelming probability, we prove that a less stringent condition is that the matrix rows are independent and uniformly sub-Gaussian. This does not impose in particular that all matrix coefficients are independent, but only their rows, which is a weaker condition.