On spectra perturbation and elementary divisors of positive matrices

Ccapa, Javier ; Soto, Ricardo L.

The electronic journal of linear algebra, Tome 18 (2009), pp. 462-481.

Résumé

Summary: A remarkable result of Guo [Linear Algebra Appl., 266:261-270, 1997] establishes that if the list of complex numbers $\Lambda = {\lambda 1 , \lambda 2 , . . . , \lambda n }$ is the spectrum of an n $\times n$ nonnegative matrix, where $\lambda 1$ is its Perron root and $\lambda 2 \in R$, then for any t > 0, the list $\Lambda t = {\lambda 1$ + t, $\lambda 2 \pm t, \lambda 3 , . . . , \lambda n }$ is also the spectrum of a nonnegative matrix. In this paper it is shown that if $\lambda 1 > \lambda 2 \geq . . . \geq \lambda n \geq 0$, then Guo's result holds for positive stochastic, positive doubly stochastic and positive symmetric matrices. Stochastic and doubly stochastic matrices are also constructed with a given spectrum and with any legitimately prescribed elementary divisors.

Détail

Classification : 15A18
Keywords: stochastic matrix, doubly stochastic matrix, symmetric matrix, spectrum perturbation, elementary divisors