The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?

Abstract
The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/||Anx|| converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated metric, which would also turn positive matrices into contractions. The main result of this paper is that, essentially, this is impossible. The paper also gives some other results of possible interest in themselves, as well as enough background to make the presentation self-contained.

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