Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem

In the special case of where the spectrum σ = {λ1, λ2, λ3, 0, 0, . . . , 0} has at most three nonzero eigenvalues λ1, λ2, λ3 with λ1 ≥ 0 ≥ λ2 ≥ λ3, and λ1 + λ2 + λ3 = 0, the inverse eigenvalue problem for symmetric stochastic n × n matrices is solved. Constructions are provided for the appropriate matrices where they are readily available. It is shown that when n is odd it is not possible to realize the spectrum σ with an n× n symmetric stochastic matrix when λ3 = 0 and 3 2n−3 > λ2 λ3 ≥ 0, and it is shown that this bound is best possible.