Premium
On computing minimal realizable spectral radii of non‐negative matrices
Author(s) -
Chu Moody T.,
Xu Shufang
Publication year - 2005
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.395
Subject(s) - spectral radius , mathematics , eigenvalues and eigenvectors , spectrum (functional analysis) , inverse , simple (philosophy) , matrix (chemical analysis) , computation , set (abstract data type) , tuple , discrete mathematics , combinatorics , algorithm , computer science , geometry , philosophy , physics , materials science , epistemology , quantum mechanics , composite material , programming language
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 = (λ 2 ,…,λ n ) ∈ ℂ n –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 {λ 1 ,…,λ n } is precisely the spectrum of a certain n × n non‐negative matrix with λ 1 as its spectral radius if and only if λ 1 ⩾ 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.