Abstract :
In this paper, we study the region of 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 .
