# On spectra perturbation and elementary divisors of positive matrices

## Main Article Content

## Abstract

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 n × n 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.

## Article Details

Issue

Section

Article