Spectral radius and infinity norm of matrices |
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Authors: | Baodong Zheng Liancheng Wang |
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Affiliation: | a Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China b Department of Mathematics and Statistics, Kennesaw State University, Kennesaw, GA 30144-5591, USA |
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Abstract: | Let Mn(R) be the linear space of all n×n matrices over the real field R. For any A∈Mn(R), let ρ(A) and ‖A‖∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φa on Mn(R), we show that, for any A∈Mn(R), ρ(A)<‖A‖∞ if . If A∈Mn(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. |
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Keywords: | Spectral radius Infinity norm Dynamic system Asymptotical stability |
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