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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The Perron-Frobenius theorem for homogeneous, monotone functions
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by Stéphane Gaubert and Jeremy Gunawardena PDF
Trans. Amer. Math. Soc. 356 (2004), 4931-4950 Request permission


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
  • Email:
  • 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
  • Email:,
  • Received by editor(s): May 10, 2001
  • Received by editor(s) in revised form: July 2, 2003
  • Published electronically: March 23, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 4931-4950
  • MSC (2000): Primary 47J10; Secondary 47H09, 47H07, 15A48
  • DOI:
  • MathSciNet review: 2084406