The Perron-Frobenius theorem for homogeneous, monotone functions

Abstract

If A A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A A has an eigenvector in the positive cone, ( R + ) n (\mathbb R^{+})^n . We associate a directed graph to any homogeneous, monotone function, f : ( R + ) n → ( R + ) n f: (\mathbb R^{+})^n \rightarrow (\mathbb R^{+})^n , and show that if the graph is strongly connected, then f f has a (nonlinear) eigenvector in ( R + ) n (\mathbb 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.