Spectral Radius and Infinity Norm of Matrices

Authors

Baodong Zheng, Harbin Institute of Technology, China
Liancheng Wang, Kennesaw State UniversityFollow

Department

Mathematics

Document Type

Article

Publication Date

10-2008

Abstract

Let M_n(R) be the linear space of all n×n matrices over the real field R. For any AM_n(R), let ρ(A) and A_∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φ_a on M_n(R), we show that, for any AM_n(R), ρ(A)<A_∞ if . If AM_n(R) is nonnegative, we prove that ρ(A)<A_∞ if and only if , and ρ(A)=A_∞ if and only if the transformation φ_A∞ preserves the spectral radius and the infinity norm of A. As an application, we investigate a class of linear discrete dynamic systems in the form of X(k+1)=AX(k). The asymptotical stability of the zero solution of the system is established by a simple algebraic method.