, Volume 96, Issue 2, pp 363-376
Date: 06 Jun 2003

On the powers of a matrix with perturbations

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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 ν k is a suitable scaling factor.

Mathematics Subject Classification (2000): 15A60, 65F15