Optimization of the spectral radius of a product for nonnegative matrices

Jonathan Axtell, Lixing Han, Daniel Hershkowitz, Michael Neumann, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)

Abstract

Let A be an n × n irreducible nonnegative matrix. We show that over the set Ωnof all n × n doubly stochastic matrices S, the multiplicative spectral radius ρ (SA) attains a minimum and a maximum at a permutation matrix. For the case when A is a symmetric nonnegative matrix, a by-product of our technique of proof yields a result allowing us to show that ρ (S1A) ≥ ρ (S2A), when S1and S2are two symmetric matrices such that both S1A and S2A are nonnegative matrices and S1- S2is a positive semidefinite matrix. This result has several corollaries. One corollary is that ρ (S1A) ≥ ρ (S2A), when S1= (1 / n) J and S2= (1 / (n - 1)) (J - I), where J is the matrix of all 1's. A second corollary is a comparison theorem for weak regular splittings of two monotone matrices.
Original languageEnglish
Pages (from-to)1442-1451
Number of pages10
JournalLinear Algebra and Its Applications
Volume430
Issue number5-6
DOIs
Publication statusPublished - 1 Mar 2009
Externally publishedYes

Keywords

  • Nonnegative matrix
  • Spectral radius
  • Stochastic matrix

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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