A metric variant of Frobenius's theorem and some other remarks on positive matrices

Abstract

The theory of positive (=nonnegative) finite square matrices continues, three quarters of a century after the pioneering and well-known papers of Perron and Frobenius [4], to present a multitude of different aspects. This is evidenced, for example, by the recent papers [1] and [2], as well as by the vast literature concerned with extensions to operators on infinite dimensional spaces (see [5]). Supposing A to be a positive n × n matrix with spectral radius r( A) = 1, the main purpose of this note is to display the role of λ = 1 as a root of the minimal polynomial of A (or equivalently, of certain norm conditions on A, for the lattice structure of the space M spanned by the unimodular eigenvectors of A as well as for the permutational character of A on M. Proposition 1 can thus be viewed as a variant of Frobenius's theorem on the peripheral spectrum of indecomposable square matrices, and we hope that the proof of Proposition 2 will clarify to what extent indecomposability is responsible for the main results available in that special case. The remaining remarks (Propositions 3 and 4) are concerned with the spectral characterization of permutation matrices and with finite groups of positive matrices. Some of that material is undoubtedly known, but we give simple, transparent proofs.