On positive stochastic matrices with real characteristic roots

Abstract

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.