Perturbing non-real eigenvalues of non-negative real matrices

Let σ=(ρ,b+ic,b-ic,λ4,…,λn) be the spectrum of an entry non-negative matrix and t⩾0. Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73–76] has shown that σ=(ρ+2t,b-t+ic,b-t-ic,λ4,…,λn) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t⩾0.