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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 =ν k P k u 1 , where ν k is a suitable scaling factor.

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Authors and Affiliations

  1. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742

    G.W. Stewart

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  1. G.W. Stewart
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Correspondence to G.W. Stewart.

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

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Stewart, G. On the powers of a matrix with perturbations. Numer. Math. 96, 363–376 (2003). https://doi.org/10.1007/s00211-003-0470-0

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  • DOI: https://doi.org/10.1007/s00211-003-0470-0

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