Lower bounds for the Perron root of a non-negative irreducible matrix

Abstract

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.