Guo perturbation for symmetric nonnegative circulant matrices

Abstract

Guo[W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270] sets the question: if the list Λ = { λ 1 , λ 2 , … , λ n } is symmetrically realizable (that is, Λ is the spectrum of a symmetric nonnegative matrix), and t > 0 , whether or not the list Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } is also symmetrically realizable. In this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant. We also give a necessary and sufficient condition for Λ to be the spectrum of a nonnegative left circulant matrix.