On spectra perturbation and elementary divisors of positive matrices

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Javier Ccapa
Ricardo L. Soto


A remarkable result of Guo [Linear Algebra Appl., 266:261–270, 1997] establishes that
if the list of complex numbers Λ = {λ1, λ2,...,λn} is the spectrum of an n × n nonnegative matrix, where λ1 is its Perron root and λ2 ∈ R, then for any t > 0, the list Λt = {λ1 +t, λ2 ±t, λ3,...,λn} is also the spectrum of a nonnegative matrix. In this paper it is shown that if λ1 > λ2 ≥ ... ≥ λn ≥ 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.

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