A note on Guo's perturbation result for symmetric nonnegative matrices
Abstract
In 1997 Guo Wuwen proved that if (λ1,λ2,…,λn) is the spectrum of a nonnegative matrix with Perron eigenvalue λ1 and λ2 real, then (λ1+t,λ2± t,λ3,…,λn) is the spectrum of a nonnegative matrix for every t≥ 0. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if A is a symmetric nonnegative matrix with eigenvalues λ1≥…≥λn and d1≥|d2|+…+|dn|, then there is a symmetric matrix B≥ A with eigenvalues λ1+d1,…,λn+dn, and B has the same diagonal as A when d1+…+dn=0. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if A is irreducible and every nonnegative symmetric matrix that commutes with A and vanishes wherever A is positive is a multiple of the identity, then the spectrum of A can be moved by a small amount in every trace-preserving direction (in every direction, if A has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.
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BibTeX
@misc{smigoc2026note,
title = {{A note on Guo's perturbation result for symmetric nonnegative matrices}},
author = {Helena Šmigoc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14759},
eprint = {2610.14759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14759},
}
RIS
TY - UNPB
TI - A note on Guo's perturbation result for symmetric nonnegative matrices
AU - Helena Šmigoc
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14759
UR - https://arxiv.org/abs/2610.14759
AB - In 1997 Guo Wuwen proved that if (λ_1,λ_2,…,λ_n) is the spectrum of a nonnegative matrix with Perron eigenvalue λ_1 and λ_2 real, then (λ_1+t,λ_2± t,λ_3,…,λ_n) is the spectrum of a nonnegative matrix for every t≥ 0. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if A is a symmetric nonnegative matrix with eigenvalues λ_1≥…≥λ_n and d_1≥|d_2|+…+|d_n|, then there is a symmetric matrix B≥ A with eigenvalues λ_1+d_1,…,λ_n+d_n, and B has the same diagonal as A when d_1+…+d_n=0. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if A is irreducible and every nonnegative symmetric matrix that commutes with A and vanishes wherever A is positive is a multiple of the identity, then the spectrum of A can be moved by a small amount in every trace-preserving direction (in every direction, if A has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.
ER -