On elementary divisors perturbation of nonnegative matrices

An outstanding result of Guo [W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261-270] establishes that if the list Λ = {λ₁, λ₂, ..., λ_n} is the spectrum of an n × n nonnegative matrix, where λ₁ is its Perron eigenvalue and λ₂ ∈ R, then for any t ≥ 0, the list Λ_t = {λ₁ + t, λ₂ ± t, ..., λ_n} is also the spectrum of a nonnegative matrix. In this paper we extend the result of Guo to elementary divisors. In particular, if A is a nonnegative matrix with spectrum Λ then, by means of two rank one perturbations, we construct a modified matrix B, which is also nonnegative, with spectrum Λ_t and we explicitly provide the Jordan canonical form of B.