Continuity of the generalized spectral radius in max algebra

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• 作者：Yung-Yih Lur ; Wen-Wei Yang
• 关键词：Max algebra ; Generalized spectral radius ; Joint spectral radius ; Simultaneous nilpotence
• 刊名：Linear Algebra and its Applications
• 出版年：2009
• 出版时间：15 April 2009
• 年：2009
• 卷：430
• 期：8-9
• 页码：2301-2311
• 全文大小：173 K

文摘

Let · be an induced matrix norm associated with a monotone norm on and β be the collection of all nonempty closed and bounded subsets of n×n nonnegative matrices under this matrix norm. For Ψ,Φβ, the Hausdorff metric H between Ψ and Φ is given by H(Ψ,Φ)=max{sup_AΨinf_BΦA-B,sup_BΦinf_AΨA-B}. The max algebra system consists of the set of nonnegative numbers with sum ab=max{a,b} and the standard product ab for a,b0. For n×n nonnegative matrices A,B their product is denoted by AB, where [AB]_ij=max_1kna_ikb_kj. For each Ψβ, the max algebra version of the generalized spectral radius of Ψ is , where . 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.