On the symmetric doubly stochastic inverse eigenvalue problem

Let σ=(1,λ2,…,λn)σ=(1,λ2,…,λn) be a list of real numbers. The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the list σ   which can occur as the spectrum of an n×nn×n symmetric doubly stochastic matrix A. If the matrix A   is positive, we can necessarily obtain a subproblem, symmetric positive doubly stochastic inverse eigenvalue problem (hereafter SPDIEP), of the SDIEP. In this paper, we give some sufficient conditions for the SDIEP and SPDIEP and prove that the set formed by the spectra of all n×nn×n symmetric positive doubly stochastic matrices is non-convex for n⩾4n⩾4.