A note on the boundary of the set where the decreasingly ordered spectra of symmetric doubly stochastic matrices lie

In this paper, we study the region of Rn where the decreasingly ordered spectra of all the n × n symmetric doubly stochastic matrices lie with emphasis on the boundary set of . As applications, we study the case n = 4 and in particular we solve the inverse eigenvalue problem for 4 × 4 symmetric doubly stochastic matrices of trace zero by using different techniques than that used in [H. Perfect, L. Mirsky, Spectral properties of doubly stochastic matrices, Monatsh. Math. 69 (1965) 35–57]. Also, we solve the same problem for 4 × 4 symmetric doubly stochastic matrices of trace two which serves only to illustrate this paper’s method. In addition, we describe a nonconvex region Ef of which corresponds to new sufficient conditions for the 4 × 4 symmetric doubly stochastic matrices. At the end, we conjecture that .