Perturbing non-real eigenvalues of nonnegative real matrices

Abstract

Let A be an (entrywise) nonnegative nxn matrix with spectrum σ and Perron eigenvalue ρ. Guo Wuwen [Linear Algebra and its Applications 266 (1997), pp. 261-267] has shown if λ is another real eigenvalue of A, then, for all t ≥ 0, replacing ρ, λ in σ by ρ+t, λ−t, respectively, while keeping all other entries of σ unchanged, again yields the spectrum of a nonnegative matrix. He poses the question of whether an analogous result holds in the case of non-real λ. In this paper, it is shown that this question has an affirmative answer.