An inverse problem for symmetric doubly stochastic matrices

Abstract

In this paper, we study the inverse eigenvalue problem for n × n symmetricdoubly stochastic matrices. The spectra of all indecomposable imprimitive symmetric doublystochastic matrices are characterized. Then we obtain new sufficient conditions for a realn-tuple to be thespectrum of an n × nsymmetric doubly stochastic matrix of zero trace. Also, we provethat the set where the decreasingly ordered spectra of all n × nsymmetric doubly stochastic matrices lie is not convex. As a consequence, weprove that the set where the decreasingly ordered spectra of all n × nnon-negative matrices lie is not convex.