The Perron-Frobenius theorem for homogeneous, monotone functions
By Stephane Gaubert, Jeremy Gunawardena. 2001
164 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/852e3ff712e6d328fac8b5eef9ccf5e824797f9f
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In Computational Theory and Mathematics
- Marianne Akian
- Stephane Gaubert
- Sergey N. Dashkovskiy
- Björn S. Rüffer
- Fabian R. Wirth
- Andrii Mironchenko
- Ricardo D. Katz
- Hiroshi Ito
In Numerical Analysis
Citations in the corpus
- Spectral theorem for convex monotone homogeneous maps, and ergodic control (2001, methodology): “tes the j-th vector of the canonical basis of Rn. Another way to guarantee the existence of an eigenvector is to use the convex spectral theorem itself, thanks to the following observation taken from [GG01]. We denote by fb(x) = limµ→∞ µ−1f(µx) the recession function of f (fbneed not exist when f is”
- Small gain theorems for large scale systems and construction of ISS Lyapunov functions (2009, background): “By the generalized Perron–Frobenius theorem [13] there exists a positive eigenvector s ∈ R+.”
- Perron–Frobenius theorem for nonnegative multilinear forms and extensions (2009, unknown): “In § 2 we recall the results of [3, 9] on existence and uniqueness of positive eigenvectors of homogeneous monotone maps that act on the interior of the cone R n + .”