A matrix realization of spectral bounds of the spectral radius of a nonnegative matrix

TL;DR

This paper introduces a matrix-based approach to derive sharp spectral bounds for the spectral radius of nonnegative matrices, providing new upper and lower bounds using eigenvalues of smaller related matrices.

Contribution

It presents a novel matrix realization method for spectral bounds, enabling easier computation of sharp bounds for the spectral radius of nonnegative matrices.

Findings

Sharp upper bound expressed by sum, largest off-diagonal, and largest diagonal entries.

New class of sharp lower bounds involving row-sums and matrix entries.

Method simplifies spectral radius estimation for nonnegative matrices.

Abstract

We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give a sharp upper bound of the spectral radius of $C$ expressed by the sum of entries, the largest off-diagonal entry $f$ and the largest diagonal entry $d$ in $C$. We also give a new class of sharp lower bounds of the spectral radius of $C$ expressed by the above $d$ and $f$, the least row-sum $r_n$ and the $t$-th largest row-sum $r_t$ in $C$ satisfying $0<r_n-(n-t-1)f-d\leq r_t-(n-t)f$, where $n$ is the size of $C$.