Perturbing non-real eigenvalues of nonnegative real matrices

Laffey, Thomas J.

The electronic journal of linear algebra, Tome 12 (2004-2005).

Résumé

Summary: Let $A$ be an (entrywise) nonnegative $n\times n$ matrix with spectrum $\sigma$ and Perron eigenvalue $\rho$. Guo Wuwen [Linear Algebra and its Applications 266 (1997), pp. 261-267] has shown that if $\lambda$ is another real eigenvalue of $A$, then, for all $t\ge 0$, replacing $\rho, \lambda$ in $\sigma$ by $\rho+t, \lambda-t$, respectively, while keeping all other entries of $\sigma$ 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 $\lambda$. In this paper, it is shown that this question has an affirmative answer.

Détail

Classification : 15A48, 15A18
Keywords: nonnegative matrix, spectrum, realizable list, perturbation, perron eigenvector