A note on the perturbation of positive matrices by normal and unitary matrices

Michael Neumann, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm sqrt(n) was sought.
Original languageEnglish
Pages (from-to)224-229
Number of pages6
JournalLinear Algebra and Its Applications
Volume428
Issue number1
DOIs
Publication statusPublished - 1 Jan 2008
Externally publishedYes

Keywords

  • Doubly stochastic matrices
  • Nonnegative matrices
  • Normal matrices
  • Real unitary matrices
  • Spectral radius

ASJC Scopus subject areas

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

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