The inverse eigenvalue problem for symmetric doubly stochastic matrices

For a positive integer n and for a real number s, let Γns 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=(λ1,λ2,…,λn)T, there exists a symmetric matrix in Γnλ1 whose spectrum is Λ.(ii) For a real n-tuple Λ=(1,λ2,…,λn)T with 1 λ2 λn, if 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.