An inverse problem for symmetric doubly stochastic matrices

In this paper, we study the inverse eigenvalue problem for n x n symmetric doubly stochastic matrices. The spectra of all indecomposable imprimitive symmetric doubly stochastic matrices are characterized. Then we obtain new sufficient conditions for a real n-tuple to be the spectrum of an n x n symmetric doubly stochastic matrix of zero trace. Also, we prove that the set where the decreasingly ordered spectra of all n x n symmetric doubly stochastic matrices lie is not convex. As a consequence, we prove that the set where the decreasingly ordered spectra of all n x n non-negative matrices lie is not convex.