The Perron-Frobenius theorem for homogeneous, monotone functions

Stéphane Gaubert; Jeremy Gunawardena

doi:10.1090/S0002-9947-04-03470-1

The Perron-Frobenius theorem for homogeneous, monotone functions

Stéphane Gaubert
, Jeremy Gunawardena

Research output: Contribution to journal › Article › peer-review

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, (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 Hubert projective metric. They lead to further existence results and open problems.

Original language	English
Pages (from-to)	4931-4950
Number of pages	20
Journal	Transactions of the American Mathematical Society
Volume	356
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9947-04-03470-1
Publication status	Published - 1 Dec 2004

Keywords

Collatz-Wielandt property
Hubert projective metric
Nonexpansive function
Nonlinear eigenvalue
Perron-Frobenius theorem
Strongly connected graph
Supereigenspace
Topical function

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10.1090/S0002-9947-04-03470-1

Cite this

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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, (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 Hubert projective metric. They lead to further existence results and open problems.",

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AU - Gaubert, Stéphane

AU - Gunawardena, Jeremy

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N2 - 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 Hubert projective metric. They lead to further existence results and open problems.

AB - 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 Hubert projective metric. They lead to further existence results and open problems.

KW - Collatz-Wielandt property

KW - Hubert projective metric

KW - Nonexpansive function

KW - Nonlinear eigenvalue

KW - Perron-Frobenius theorem

KW - Strongly connected graph

KW - Supereigenspace

KW - Topical function

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DO - 10.1090/S0002-9947-04-03470-1

M3 - Article

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SN - 0002-9947

VL - 356

SP - 4931

EP - 4950

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

ER -

The Perron-Frobenius theorem for homogeneous, monotone functions

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Keywords

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