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The Perron-Frobenius theorem for homogeneous, monotone functions

Authors: Stéphane Gaubert and Jeremy Gunawardena
Journal: Trans. Amer. Math. Soc. 356 (2004), 4931-4950
MSC (2000): Primary 47J10; Secondary 47H09, 47H07, 15A48
Published electronically: March 23, 2004
MathSciNet review: 2084406
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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.

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Additional Information

Stéphane Gaubert
Affiliation: INRIA, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cédex, France

Jeremy Gunawardena
Affiliation: Bauer Center for Genomics Research, Harvard University, 7 Divinity Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Systems Biology, Harvard Medical School, 200 Longwood Avenue, Boston, Massachusetts 02115

Keywords: Collatz-Wielandt property, Hilbert projective metric, nonexpansive function, nonlinear eigenvalue, Perron-Frobenius theorem, strongly connected graph, super-eigenspace, topical function
Received by editor(s): May 10, 2001
Received by editor(s) in revised form: July 2, 2003
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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