The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

Details The Perron-Frobenius Theorem for Homogeneous, Monotone Functions The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

2001-05-11

arxiv.org/abs/math/0105091v2

Trans. Amer. Math. Soc. 356 (2004), 4931-4950.

Mathematics

Functional Analysis

20 pages, 3 Postscript figures, v2 (minor revision)

Scientific paper

If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R^+)^n. We associate a directed graph to any homogeneous, monotone function, f: (R^+)^n -> (R^+)^n, and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R^+)^n. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.

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