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Distance Evaluation to the Set of Matrices with Multiple Eigenvalues

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Computer Algebra in Scientific Computing (CASC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13366))

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Abstract

The problem of finding the Frobenius distance in the \(\mathbb R^{n\times n} \) matrix space from a given matrix to the set of matrices possessing multiple eigenvalues is considered. Two approaches are discussed: the one is reducing the problem to a constrained optimization problem in \(\mathbb R^n\) with a quartic objective function, and the other one is connected with the singular value analysis for an appropriate matrix in \(\mathbb R^{2n\times 2n} \). Several examples are presented including classes of matrices where the distance in question can be explicitly expressed via the matrix eigenvalues.

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Acknowledgments

The authors are grateful to Prof. Evgenii V. Vorozhtsov and to the anonymous referees for valuable suggestions that helped to improve the quality of the paper.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, St. Petersburg State University, 7–9 Universitetskaya nab., St. Petersburg, 199034, Russia

    Elizaveta Kalinina & Alexei Uteshev

Authors
  1. Elizaveta Kalinina
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  2. Alexei Uteshev
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Correspondence to Elizaveta Kalinina .

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Editors and Affiliations

  1. Université de Lille, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Kalinina, E., Uteshev, A. (2022). Distance Evaluation to the Set of Matrices with Multiple Eigenvalues. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_12

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  • DOI: https://doi.org/10.1007/978-3-031-14788-3_12

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  • Online ISBN: 978-3-031-14788-3

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