The inverse eigenvalue problem for symmetric doubly stochastic matrices

Abstract

For a positive integer n and for a real number s, let Γ ns denote the set of all n × n real matrices whose rows and columns have sum s. In this note, by an explicit constructive method, we prove the following. (i) Given any real n-tuple Λ = (λ 1, λ2, ..., λn)T, there exists a symmetric matrix in Γnλ1 whose spectrum is Λ. (ii) For a real n-tuple Λ = (1, λ 2, ..., λn)T with 1 ≥ λ 2 ≥ ... ≥ λn, if 1/n + λ 2/n(n-1) + λ3/n(n-2) + ... + λ n/2·1 ≥ 0, then there exists a symmetric doubly stochastic matrix whose spectrum is Λ. The second assertion enables us to show that for any λ2, ..., λn ∈ [-1/(n - 1), 1], there is a symmetric doubly stochastic matrix whose spectrum is (1, λ2, ..., λn)T and also that any number β ∈ (-1,1] is an eigenvalue of a symmetric positive doubly stochastic matrix of any order. © 2003 Elsevier Inc. All rights reserved.