SpringerLink
Log in

On the Distance to the Nearest Defective Matrix

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14139))

Included in the following conference series:

  • 228 Accesses

Abstract

The problem of finding the Frobenius distance in the \(\mathbb C^{n\times n} \) matrix space from a given matrix to the set of matrices with multiple eigenvalues is considered. The problem is reduced to the univariate algebraic equation construction via computing the discriminant of an appropriate bivariate polynomial. Several examples are presented including the cases of complex and real matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    All the decimals in the following approximation are error-free.

References

  1. Akinola, R.O., Freitag, M.A., Spence, A.: The calculation of the distance to a nearby defective matrix. Numer. Linear Algebra Appl. 21(3), 403–414 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alam, R., Bora, S.: On sensitivity of eigenvalues and eigendecompositions of matrices. Linear Algebra Appl. 396, 273–301 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alam, R., Bora, S., Byers, R., Overton, M.L.: Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components. Linear Algebra Appl. 435, 494–513 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Armentia, G., Gracia, J.-M., Velasco, F.-E.: Nearest matrix with a prescribed eigenvalue of bounded multiplicities. Linear Algebra Appl. 592, 188–209 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bikker, P., Uteshev, A.Y.: On the Bezout construction of the resultant. J. Symb. Comput. 28(1), 45–88 (1999)

    Article  MATH  Google Scholar 

  6. Demmel, J.W.: Computing stable eigendecompositions of matrices. Linear Algebra Appl. 79, 163–193 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demmel, J.W.: On condition numbers and the distance to the nearest ill-posed problem. Numer. Math. 51, 251–289 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218 (1936)

    Article  MATH  Google Scholar 

  9. Higham, N.G.: Matrix nearness problems and applications. In: Applications of matrix theory, pp. 1–27. Oxford Univ. Press, New York (1989)

    Google Scholar 

  10. Horn, R.A., Johnson, Ch.: Matrix Analysis, 2nd edn. Cambridge University Press, New York (2013)

    MATH  Google Scholar 

  11. Kalinina, E.A., Smol’kin, Y.A., Uteshev, A.Y.: Stability and distance to instability for polynomial matrix families. Complex perturbations. Linear Multilinear Algebra 70, 1291–1314 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kalinina, E., Uteshev, A.: Distance evaluation to the set of matrices with multiple eigenvalues. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2022. Lecture Notes in Computer Science, vol. 13366, pp. 206–224. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-14788-3_12

    Chapter  Google Scholar 

  13. Lippert, R.A., Edelman, A.: The computation and sensitivity of double eigenvalues. In: Chen, Z., Li, Y., Micchelli, C.A., Xu, Y. (eds.) Proceedings of the Advances in Computational Mathematics, pp. 353–393. Gaungzhou International Symposium, Dekker, New York (1999)

    Google Scholar 

  14. Malyshev, A.: A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues. Numer. Math. 83, 443–454 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mengi, E.: Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity. Numer. Math. 118, 109–135 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Oliveira, O.: The implicit and inverse function theorems: easy proofs. Real Anal. Exchange 39(1), 207–218 (2013/2014)

    Google Scholar 

  17. Petkov, P.H., Konstantinov, M.M.: The numerical Jordan form. Linear Algebra Appl. 638, 1–45 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ruhe, A.: Properties of a matrix with a very ill-conditioned eigenproblem. Numer. Math. 15, 57–60 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, New York (1965)

    MATH  Google Scholar 

  20. Wilkinson, J.H.: Note on matrices with a very ill-conditioned eigenproblem. Numer. Math. 19, 176–178 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wilkinson, J.H.: On neighbouring matrices with quadratic elementary divisors. Numer. Math. 44, 1–21 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wilkinson, J.H.: Sensitivity of eigenvalues. Util. Math. 25, 5–76 (1984)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

This research was supported by the St. Petersburg State University (project ID 96291288).

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

Author information

Authors and Affiliations

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

    Elizaveta Kalinina, Alexei Uteshev, Marina Goncharova & Elena Lezhnina

Authors
  1. Elizaveta Kalinina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexei Uteshev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marina Goncharova
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elena Lezhnina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elizaveta Kalinina .

Editor information

Editors and Affiliations

  1. University of Lille, CRIStAL, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

  4. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

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

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kalinina, E., Uteshev, A., Goncharova, M., Lezhnina, E. (2023). On the Distance to the Nearest Defective Matrix. In: Boulier, F., England, M., Kotsireas, I., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2023. Lecture Notes in Computer Science, vol 14139. Springer, Cham. https://doi.org/10.1007/978-3-031-41724-5_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-41724-5_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-41723-8

  • Online ISBN: 978-3-031-41724-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation