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Stability in Matrix Analysis Problems

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Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Abstract

Let there be given a class of matrices \(\mathcal{A}\), and for each \(X \in \mathcal{A}\), a set \(\mathcal{G}(X)\). The following two broadly formulated problems are addressed: 1. An element \(Y _{0} \in \mathcal{G}(X_{0})\) will be called stable with respect to \(\mathcal{A}\) and the collection \(\{\mathcal{G}(X)\}_{X\in \mathcal{A}}\), if for every \(X \in \mathcal{A}\) which is sufficiently close to X 0 there exists an element \(Y \in \mathcal{G}(X)\) that is as close to Y 0 as we wish. Give criteria for existence of a stable Y 0, and describe all of them. 2. Fix α ≥ 1. An element \(Y _{0} \in \mathcal{G}(X_{0})\) will be called α-stable with respect to \(\mathcal{A}\) and the collection \(\{\mathcal{G}(X)\}_{X\in \mathcal{A}}\), if for every \(X \in \mathcal{A}\) which is sufficiently close to X 0 there exists an element \(Y \in \mathcal{G}(X)\) such that the distance between Y and Y 0 is bounded above by a constant times the distance between X and X 0 raised to the power 1∕α where the constant does not depend on X. Give criteria for existence of an α-stable Y 0, and describe all of them. We present an overview of several basic results and literature guide concerning various stability notions, including the concept of conditional stability, as well as prove several new results and state open problems of interest. Large part of the work leading to this chapter was done while the second author visited Vrije Universiteit, Amsterdam, whose hospitality is gratefully acknowledged.

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Acknowledgements

We thank C. Mehl for very careful reading of the chapter and many useful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    André C. M. Ran

  2. Unit for BMI, North-West University, Potchefstroom, South Africa

    André C. M. Ran

  3. Department of Mathematics, College of William and Mary, 8795, Williamsburg, VA, 23187-8795, USA

    Leiba Rodman

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  1. André C. M. Ran
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Correspondence to André C. M. Ran .

Editor information

Editors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Carl-Friedrich-Gauß Fakultät, TU Braunschweig, Institute Computational Mathematics, Braunschweig, Germany

    Matthias Bollhöfer

  3. MATHICSE, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Daniel Kressner

  4. Institute of Mathematics, T U Berlin, Berlin, Germany

    Christian Mehl

  5. Institute of Mathematics, University of Augsburg, Augsburg, Germany

    Tatjana Stykel

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Ran, A.C.M., Rodman, L. (2015). Stability in Matrix Analysis Problems. In: Benner, P., Bollhöfer, M., Kressner, D., Mehl, C., Stykel, T. (eds) Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-15260-8_13

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