Perron-Frobenius theory of seminorms: A topological approach

Assaf Goldberger, Michael Neumann*

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

3 Scopus citations

Abstract

For nonnegative matrices A, the well known Perron-Frobenius theory studies the spectral radius ρ(A). Rump has offered a way to generalize the theory to arbitrary complex matrices. He replaced the usual eigenvalue problem with the equation |Ax| = λ|x| and he replaced ρ(A) by the signed spectral radius, which is the maximum λ that admits a nontrivial solution to that equation. We generalize this notion by replacing the linear transformation A by a map f:Cn→R whose coordinates are seminorms, and we use the same definition of Rump for the signed spectral radius. Many of the features of the Perron-Frobenius theory remain true in this setting. At the center of our discussion there is an alternative theorem relating the inequalities f(x) ≥ λ|x| and f(x) < λ|x|, which follows from topological principals. This enables us to free the theory from matrix theoretic considerations and discuss it in the generality of seminorms. Some consequences for P-matrices and D-stable matrices are discussed.

Original languageEnglish
Pages (from-to)245-284
Number of pages40
JournalLinear Algebra and Its Applications
Volume399
Issue number1-3
DOIs
StatePublished - 1 Apr 2005
EventInternational Meeting on Matrix Analysis and Applications - Ft. Lauderdale, Fl, United States
Duration: 14 Dec 200316 Dec 2003

Funding

FundersFunder number
National Science Foundation
Directorate for Mathematical and Physical Sciences0201333

    Keywords

    • Nonnegative matrices
    • P-matrices
    • Seminorms
    • Theorems on alternative

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