Spectral Radius and Infinity Norm of Matrices



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Let Mn(R) be the linear space of all n×n matrices over the real field R. For any AMn(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 AMn(R), ρ(A)<A if . If AMn(R) is nonnegative, we prove that ρ(A)<A if and only if , and ρ(A)=A if and only if the transformation φA 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.