2001/05/31 by Stephane Gaubert, Jeremy Gunawardena · 1 citation
Mathematics · #math.FA #msc:47J10 #msc:47H09 #msc:47H07 #msc:15A48
published as Trans. Amer. Math. Soc. 356 (2004), 4931-4950. · 20 pages, 3 Postscript figures, v2 (minor revision)
arxiv created 2003/08/18 · arxiv updated 2009/11/30
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.