On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral

Abstract

We first derive the bound |det(λI - A)|≤λk - λk0 (λ0≤λ), where A is a k × k nonnegative real matrix and λ0 is the spectral radius of A. If A is irreducible and integral, and its largest nonnegative eigenvalue is an integer n, then we use this inequality to derive the upper bound nk-1 on the components of the smallest integer eigenvector corresponding to n. Finer information on the components is also derived. © 1987.