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

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, ihen there exî 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.