Affordable Access

Publisher Website

Continuity of the generalized spectral radius in max algebra

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
430
Identifiers
DOI: 10.1016/j.laa.2008.12.007
Keywords
  • Max Algebra
  • Generalized Spectral Radius
  • Joint Spectral Radius
  • Simultaneous Nilpotence
Disciplines
  • Mathematics

Abstract

Abstract Let ‖ · ‖ be an induced matrix norm associated with a monotone norm on R n and β be the collection of all nonempty closed and bounded subsets of n × n nonnegative matrices under this matrix norm. For Ψ , Φ ∈ β , the Hausdorff metric H between Ψ and Φ is given by H ( Ψ , Φ ) = max { sup A ∈ Ψ inf B ∈ Φ ‖ A - B ‖ , sup B ∈ Φ inf A ∈ Ψ ‖ A - B ‖ } . The max algebra system consists of the set of nonnegative numbers with sum a ⊗ b = max { a , b } and the standard product ab for a , b ⩾ 0 . For n × n nonnegative matrices A , B their product is denoted by A ⊗ B , where [ A ⊗ B ] ij = max 1 ⩽ k ⩽ n a ik b kj . For each Ψ ∈ β , the max algebra version of the generalized spectral radius of Ψ is μ ( Ψ ) = limsup m → ∞ [ sup A ∈ Ψ ⊗ m μ ( A ) ] 1 m , where Ψ ⊗ m = { A 1 ⊗ A 2 ⊗ ⋯ ⊗ A m : A i ∈ Ψ } . Here μ ( A ) is the maximum circuit geometric mean. In this paper, we prove that the max algebra version of the generalized spectral radius is continuous on the Hausdorff metric space ( β , H ) . The notion of the max algebra version of simultaneous nilpotence of matrices is also proposed. Necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well.

There are no comments yet on this publication. Be the first to share your thoughts.

Statistics

Seen <100 times
0 Comments

More articles like this

On the continuity of the generalized spectral radi...

on Linear Algebra and its Applica... Jan 01, 2011

Generalized spectral radius and its max algebra ve...

on Linear Algebra and its Applica... Aug 15, 2013

A max version of the generalized spectral radius t...

on Linear Algebra and its Applica... Jan 01, 2006
More articles like this..