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

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.