A simple observation on random matrices with continuous diagonal entries

TL;DR

This paper establishes a probabilistic bound on the determinant of a matrix perturbed by a random diagonal matrix with continuous entries, highlighting a simple yet general property of such random matrices.

Contribution

It provides a new probabilistic bound on the determinant of matrices with independent continuous diagonal entries, extending understanding of their spectral properties.

Findings

The probability that the determinant's nth root is below a threshold t is bounded by 2bnt.

The result applies to any fixed matrix A, regardless of its structure.

The bound depends on the uniform upper bound of the densities of the diagonal entries.

Abstract

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$ \p\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt, $$ where $b>0$ is a uniform upper bound on the densities of $T_{i,i}$.