Perturbing non-real eigenvalues of non-negative real matrices

Let (sigma = (rho, b + ic, b - ic, lambda(4),....,lambda(n)) 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 (7 = (rho + 2t, b - t + ic, b - t - ic, lambda(4),...,lambda(n)) is also the spectrum of some non-negative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all non-negative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t >= 0. Crown copyright (C) 2007 Published by Elsevier Inc. All rights reserved.