l'niiii',1 III U.S.A. THE CONTRACTION MAPPING APPROACH TO THE PERRON-FROBENIUS THEORY; WHY HTLBERT'S

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, ihen there exi ̂ sts an.v,, such thai A ".\/\\A "x\ \ converges to x,j for all.v> 0, There are many classical proofs of this theorem, all depending on a connection between positivily of a matrix and properties of ils eigenvalues. A more modern proof, due to Garrett Birkhoff. is based on the observation that every linear transfornKition with a positive matrix may be viewed as a contraction mapping on the nonnegative orthanl. This observation turns the Perron-Fro ben ills theorem into a special case 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 thai positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is ralher 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, ihis 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. 1. Introduction. Th