Continuity of the generalized spectral radius in max algebra

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.