Matrices Leaving a Cone Invariant

Abstract

Hans Schneider and Bit-Shun Tam Generalizations of the Perron-Frobenius theory of nonnegative matrices to linear operators leaving a cone invariant were first developed for operators on a Banach space by Krein and Rutman [KR48], Karlin [Kar59] and Schaefer [Sfr66] although there are early examples in finite dimensions, e.g. [Sch65] and [Bir67]. In this article we describe a generalization, sometimes called the geometric spectral theory of nonnegative linear operators in finite dimensions, which emerged in the late 1980s. Motivated by a search for geometric analogs of results in the previously developed combinatorial spectral theory of (reducible) nonnegative matrices (for reviews see [Sch86] and [Her99]), this area is a study of the PerronFrobenius theory of a nonnegative matrix and its generalizations from the conetheoretic viewpoint. The treatment is linear-algebraic and cone-theoretic (geometric) with the facial and duality concepts and occasionally certain elementary analytic tools playing the dominant role. The theory is particularly rich when the underlying cone is polyhedral (finitely generated) and it reduces to the nonnegative matrix case when the cone is simplicial.