# The Perron-Frobenius theorem for homogeneous, monotone functions

```@article{Gaubert2001ThePT,
title={The Perron-Frobenius theorem for homogeneous, monotone functions},
author={St{\'e}phane Gaubert and Jeremy Gunawardena},
journal={Transactions of the American Mathematical Society},
year={2001},
volume={356},
pages={4931-4950}
}```
• Published 11 May 2001
• Mathematics
• Transactions of the American Mathematical Society
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…
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