The spread of the spectrum of a nonnegative matrix with a zero diagonal element

TL;DR

This paper establishes lower bounds for the spectral spread of nonnegative matrices with a zero diagonal element, providing optimal bounds in the case of matrices with only two eigenvalues.

Contribution

It introduces new lower bounds for the spectral spread of such matrices and proves their optimality, expanding understanding of eigenvalue distributions.

Findings

Lower bounds for the spread of matrices with zero diagonal

Optimality of the derived bounds in specific cases

Relation between spread and spectral radius

Abstract

Let $A = [a_{i j}]_{i,j=1}^n$ be a nonnegative matrix with $a_{1 1} = 0$. We prove some lower bounds for the spread $s(A)$ of $A$ that is defined as the maximum distance between any two eigenvalues of $A$. If $A$ has only two distinct eigenvalues, then $s(A) \ge \frac{n}{2(n-1)} \, r(A)$, where $r(A)$ is the spectral radius of $A$. Moreover, this lower bound is the best possible.