The Perron-Frobenius theorem for homogeneous, monotone functions

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, $(\mathbb R^{+})^n$. We associate a directed graph to any homogeneous, monotone function, $f: (\mathbb R^{+})^n \rightarrow (\mathbb R^{+})^n$, and show that if the graph is strongly connected, then $f$ has a (nonlinear) eigenvector in $(\mathbb R^{+})^n$. 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.