On Bounded Matrices with Non-Negative Elements

It is known (Perron (10); Frobenius (5, 6)) that if A = (a ik ) is a finite matrix with elements aik ⩾ 0, then A has a real, nonnegative eigenvalue μ, satisfying μ =max|λ| where λ is in the spectrum of A, with a corresponding eigenvector x = (x 1, … , xn) for which x i≥ 0. Moreover if a ik > 0, then μ is a simple point of the spectrum with an eigenvector x (unique, except for constant multiples) with components xi ≥0. Much has been written on this and related issues; cf., for example, the recent papers (4, 12) wherein are given several references.