Some results on the symmetric doubly stochastic inverse eigenvalue problem

The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an n-tuple σ = (1; λ2; λ3; ...,λn) Є R^n with |λi| ≤ 1; i = 1; 2; ..., n, to be the spectrum of an nxn symmetric doubly stochastic matrix A. If there exists an nxn symmetric doubly stochastic matrix A with σ as its spectrum, then the list σ is s.d.s. realizable, or such that A s.d.s. realizes σ. In this paper, we propose a new sufficient condition for the existence of the symmetric doubly stochastic matrices with prescribed spectrum. Finally, some results about how to construct new s.d.s. realizable lists from the known lists are presented.