Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem
Robert Reams · Electronic Journal of Linear Algebra · 2002
In the special case of where the spectrum σ = {λ 1 , λ 2 , λ 3 , 0, 0, . . ., 0} has at most three nonzero eigenvalues λ 1 , λ 2 , λ 3 with λ 1 ≥ 0 ≥ λ 2 ≥ λ 3 , and λ 1 + λ 2 + λ 3 = 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 λ 3 = 0 and 3 2n-3 > λ 2 λ 3 ≥ 0, and it is shown that this bound is best possible.