On spectra perturbation and elementary divisors of positive matrices

A remarkable result of Guo [Linear Algebra Appl., 266:261-270, 1997] establishes that if the list of complex numbers λ = {λ ₁, λ ₂, ..., λ _n} is the spectrum of an nxn nonnegative matrix, where λ ₁ is its Perron root and λ ₂ ∈ ℝ, then for any t > 0, the list λ _t = {λ ₁+t, λ ₂± t, λ ₃, ..., λ _n} is also the spectrum of a nonnegative matrix. In this paper it is shown that if λ ₁ > λ ₂ ≥...≥ λ _n ≥ 0, then Guo's result holds for positive stochastic, positive doubly stochastic and positive symmetric matrices. Stochastic and doubly stochastic matrices are also constructed with a given spectrum and with any legitimately prescribed elementary divisors.