ON SOME USEFUL PROPERTIES OF THE PERRON EIGENVALUE OF A POSITIVE RECIPROCAL MATRIX IN THE CONTEXT OF THE ANALYTIC HIERARCHY PROCESS

A positive n X n matrix, R = (r(ij)), is said to be reciprocal if its entries verify r(ji) = 1/r(ij) > 0 for all 1 less-than-or-equal-to i,j less-than-or-equal-to n. In the context of the analytic hierarchy pro cess, where such matrices arise from the pairwise comparison of n grea ter-than-or-equal-to 2 decision alternatives on an arbitrary ratio sca le, Saaty (in Journal of Mathematical Psychology, 1977) proposed to us e mu = (lambda(max) - n)/(n - 1) greater-than-or-equal-to 0, a linear transform of the Perron eigenvalue lambda(max) of R, as a measure of t he cardinal consistency in an agent's responses and posed the problem of determining how it might vary as a function of the r(ij)'s. He also suggested that an upper bound could be found for that consistency ind ex when the entries of R are restricted to take their values in a boun ded set. Both of these questions are answered here using classical res ults from linear algebra.