The inverse eigenvalue problem for symmetric doubly stochastic matrices

For a positive integer n and for a real number s, let Gamma(n)(s), denote the set of all n x 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 Lambda = (lambda(1), lambda(2),...lambda(n))(T), there exists a symmetric matrix in Gamma(n)(lambda1) whose spectrum is Lambda. (ii) For a real n-tuple Lambda = (1, lambda(2),...lambda(n))(T) with 1 greater than or equal to lambda(2) greater than or equal to lambda(n), if 1/n + lambda(2)/n(n-1) + lambda(3)/n(n-2) + ... + lambda(n)/2(.)1 greater than or equal to 0, then there exists a symmetric doubly stochastic matrix whose spectrum is Lambda. The second assertion enables us to show that for any lambda(2),...,lambda(n) is an element of [-1/(n-1), 1], there is a symmetric doubly stochastic matrix whose spectrum is (1, lambda(2),...,lambda(n))(T) and also that any number beta is an element of (- 1, 1] is an eigenvalue of a symmetric positive doubly stochastic matrix of any order. (C) 2003 Elsevier Inc. All rights reserved.