On the powers of a matrix with perturbations

Let A be a matrix of order n. The properties of the powers A k of A have been extensively studied in the literature. This paper concerns the perturbed powers $${{ P_{{k}} = (A+E_{{k}})(A+E_{{k-1}})\\cdots(A+E_{{1}}), }}$$ where the E k are perturbation matrices. We will treat three problems concerning the asymptotic behavior of the perturbed powers. First, determine conditions under which ${{P_{{k}}\\rightarrow 0}}$ . Second, determine the limiting structure of P k . Third, investigate the convergence of the power method with error: that is, given u 1 , determine the behavior of u k =¿ k P k u 1 , where ¿ k is a suitable scaling factor.