Comment on “A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices”

In [S.K. Hwang, S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77-83] it was claimed that: if 1>λ2⩾λ3⩾⋯⩾λn and 1n+λ2n(n−1)+λ3(n−1)(n−2)+⋯+λn2⋅1⩾0, then there is a symmetric positive doubly stochastic matrix A with the eigenvalues 1,λ2,λ3,…,λn. Afterwards, Fang [M.Z. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 432 (2010) 2925-2927] presented a counterexample to demonstrate that the above proposition was inaccurate. However, the author did not give a solution for a real n-tuple σ=(1,λ2,λ3,…,λn) to be the spectrum of a symmetric positive doubly stochastic matrix of order n. In this paper, we give some sufficient conditions to make up for this deficiency.