HP Labs Technical Reports
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Existence of Eigenvectors for Monotone Homogeneous Functions
Keyword(s): Collatz-Wielandt property; Hilbert projective metric; nonexpansive function; nonlinear eigenvalue; Perron- Frobenius theorem; strongly connected graph; sub- eigenspace
Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. We consider function f : Rn Rn which are additively homogeneous and monotone in the product ordering on Rn (topical functions). We show that if some non-empty sub- eigenspace of f is bounded in the Hilbert semi-norm then f has an additive eigenvector and we give a Collatz-Wielandt characterisation of the corresponding eigenvalue. The boundedness condition is satisfied if a certain directed graph associated to f is strongly connected. The Perron-Frobenius theorem for non- negative matrices, its analogue for the max-plus semiring, a version of the mean ergodic theorem for Markov chains and theorems of bather and Zijm all follow as immediate corollaries.
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