Spectral radius and infinity norm of matrices

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 ϕn A ∞ (A) = 0. If A ∈ Mn(R) is nonnegative, we prove that ρ(A) < A ∞ if and only if ϕn A ∞ (A) = 0, 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