Stewart, G. Numer. Math. (2003) 96: 363. doi:10.1007/s00211-003-0470-0

Summary.

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}=ν_{k}P_{k}u_{1}, where ν_{k} is a suitable scaling factor.