The Perron-Frobenius Theorem for Homogeneous, Monotone Functions
Stephane Gaubert, Jeremy Gunawardena
Abstract
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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