On computing minimal realizable spectral radii of non-negative matrices

Abstract

For decades considerable efforts have been exerted to resolve the inverse eigenvalue problem for non-negative matrices. Yet fundamental issues such as the theory of existence and the practice of computation remain open. Recently, it has been proved that, given an arbitrary (n–1)-tuple ℒ = (λ2,…,λn) ∈ ℂn–1 whose components are closed under complex conjugation, there exists a unique positive real number ℛ(ℒ), called the minimal realizable spectral radius of ℒ, such that the set {λ1,…,λn} is precisely the spectrum of a certain n × n non-negative matrix with λ1 as its spectral radius if and only if λ1 ⩾ ℛ(ℒ). Employing any existing necessary conditions as a mode of checking criteria, this paper proposes a simple bisection procedure to approximate the location of ℛ(ℒ). As an immediate application, it offers a quick numerical way to check whether a given n-tuple could be the spectrum of a certain non-negative matrix. Copyright © 2004 John Wiley & Sons, Ltd.