Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

Springer Nature Link
Log in

Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev’s formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Bunse-Gerstner A., Byers R., Mehrmann V., Nichols N.K.: Numerical computation of an analytic singular value decomposition of a matrix valued function. Numer. Math. 60, 1–39 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Byers, R.: Detecting nearly uncontrollable pairs. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds.) Proceedings of the International Symposium MTNS-89, vol. III. North-Holland, Amsterdam (1990)

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Gablonsky, J.M.: Modifications of the DIRECT algorithm. Ph.D. thesis. North Carolina State University, Raleigh, North Carolina (2001)

  6. Golub G., Van Loan C.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  7. Hinrichsen, D., Motscha, M.: Optimization problems in the robustness analysis of linear state space systems. In: Proceedings of the Seminar on Approximations and Optimization, Havana, LN Mathematics. Springer, Berlin (1987)

  8. Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, London (1985)

    MATH  Google Scholar 

  9. Jones D.R., Perttunen C.D., Stuckman B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kato T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    MATH  Google Scholar 

  11. Lancaster P., Tismenetsky M.: The Theory of Matrices, 2nd edn. Academic Press, London (1985)

    MATH  Google Scholar 

  12. Lehoucq, R., Sorensen, D., Yang, C.: ARPACK users’ guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods (1997)

  13. Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(3, (Ser. B)), 503–528 (1989)

  14. Van Loan C.F.: How near is a stable matrix to an unstable matrix. Contemp. Math. 47, 465–477 (1985)

    MathSciNet  Google Scholar 

  15. Malyshev A.N.: 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  MATH  MathSciNet  Google Scholar 

  16. Paige C.C.: Properties of numerical algorithms related to computing controllability. IEEE Trans. Automat. Contr. 26(1), 130–138 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Piyavskii S.A.: An algorithm for finding the absolute extremum of a function. USSR Comput. Math. Math. Phys. 12, 57–67 (1972)

    Article  Google Scholar 

  18. Qiu L., Bernhardsson B., Rantzer A., Davison E.J., Young P.M., Doyle J.C.: A formula for computation of the real stability radius. Automatica 31(6), 879–890 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rellich F.: Storungstheorie der spektralzerlegung, I. Math. Ann. 113, 600–619 (1937)

    Article  MathSciNet  Google Scholar 

  20. Rellich F.: Storungstheorie der spektralzerlegung, II. Math. Ann. 113, 677–685 (1937)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rellich F.: Storungstheorie der spektralzerlegung, III. Math. Ann. 116, 555–570 (1939)

    Article  MathSciNet  Google Scholar 

  22. Rellich F.: Storungstheorie der spektralzerlegung, IV. Math. Ann. 117, 356–382 (1940)

    Article  MathSciNet  Google Scholar 

  23. Rellich F.: Storungstheorie der spektralzerlegung, V. Math. Ann. 118, 462–484 (1942)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Sergeyev Y.D., Kvasov D.E.: Global search based on efficient diagonal partitions and a set of lipschitz constants. SIAM J. Optim. 16(3), 910–937 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shubert B.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9, 379–388 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wilkinson J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Koc University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey

    Emre Mengi

Authors
  1. Emre Mengi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emre Mengi.

Additional information

The computing resources for this work were supplied through the National Science Foundation Grants DMS-0715146 and DMS-0821816. This work was also supported in part by the TUBITAK (the scientific and technological research council of Turkey) Grant 109T660. Most of this work was completed when the author was holding a S.E.W. assistant professorship in the department of mathematics at the University of California, San Diego.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mengi, E. Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity. Numer. Math. 118, 109–135 (2011). https://doi.org/10.1007/s00211-010-0326-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-010-0326-3

Mathematics Subject Classification (2000)

Search

Navigation