The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

TL;DR

This paper extends the Perron-Frobenius theorem to homogeneous, monotone functions by associating a directed graph and demonstrating the existence of eigenvectors under strong connectivity, with implications for invariant set boundedness.

Contribution

It introduces a graph-based approach to establish eigenvector existence for nonlinear functions, generalizing classical linear results to a broader class of functions.

Findings

Eigenvectors exist for strongly connected associated graphs.

Results unify and extend previous theorems in the literature.

New insights into invariant set boundedness in Hilbert metric.

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.