Skip to main content
SpringerLink
Log in

Structured Eigenvalue Perturbation Theory

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

Abstract

We give an overview of Volker Mehrmann’s work on structured perturbation theory of eigenvalues. In particular, we review his contributions on perturbations of structured pencils arising in control theory and of Hamiltonian matrices. We also give a brief outline of his work on structured rank one perturbations.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Adhikari, B., Alam, R.: Structured backward errors and pseudospectra of structured matrix pencils. SIAM J. Matrix Anal. Appl. 31, 331–359 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adhikari, B., Alam, R., Kressner, D.: Structured eigenvalue condition numbers and linearizations for matrix polynomials. Linear Algebra. Appl. 435, 2193–2221 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahmad, Sk.S., Mehrmann, V.: Perturbation analysis for complex symmetric, skew symmetric even and odd matrix polynomials. Electron. Trans. Numer. Anal. 38, 275–302 (2011)

    Google Scholar 

  4. Ahmad, Sk.S., Mehrmann, V.: Backward errors for Hermitian, skew-Hermitian, H-even and H-odd matrix polynomials. Linear Multilinear Algebra 61, 1244–1266 (2013)

    Google Scholar 

  5. Alam, R., Bora, S., Karow, M., Mehrmann, V., Moro, J.: Perturbation theory for Hamiltonian matrices and the distance to bounded realness. SIAM J. Matrix Anal. Appl. 32, 484–514 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Al-Ammari, M.: Analysis of structured polynomial eigenvalue problems. PhD thesis, School of Mathematics, University of Manchester, Manchester (2011)

    Google Scholar 

  7. Al-Ammari, M., Tisseur, F.: Hermitian matrix polynomials with real eigenvalues of definite type. Part 1 Class. Linear Algebra Appl. 436, 3954–3973 (2012)

    Google Scholar 

  8. Antoulas, T: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  9. Ballantine, C.S.: Numerical range of a matrix: some effective criteria. Linear Algebra Appl. 19, 117–188 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benner, P., Mehrmann, V., Xu, H.: Perturbation analysis for the eigenvalue problem of a formal product of matrices. BIT Numer. Math. 42, 1–43 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Benner, P., Kressner, D., Mehrmann, V.: Structure preservation: a challenge in computational control. Future Gener. Comput. Syst. 19, 1243–1252 (2003)

    Article  Google Scholar 

  12. Benner, P., Byers, R., Mehrmann, V., Xu, H.: Robust method for robust control. Preprint 2004–2006, Institut für Mathematik, TU Berlin (2004)

    Google Scholar 

  13. Benner, P., Kressner, D., Mehrmann, V.: Skew-Hamiltonian and Hamiltonian eigenvalue problems: theory, algorithms and applications. In: Drmac, Z., Marusic, M., Tutek, Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, Brijuni, pp. 3–39. Springer (2005). [ISBN:1-4020-3196-3]

    Google Scholar 

  14. Bora, S.: Structured eigenvalue condition number and backward error of a class of polynomial eigenvalue problems. SIAM J. Matrix Anal. Appl. 31, 900–917 (2009)

    Article  MathSciNet  Google Scholar 

  15. Bora, S., Mehrmann, V.: Linear perturbation theory for structured matrix pencils arising in control theory. SIAM J. Matrix Anal. Appl. 28, 148–191 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bora, S., Srivastava, R.: Distance problems for Hermitian matrix pencils with eigenvalues of definite type (submitted)

    Google Scholar 

  17. Brüll, T., Schröder, C.: Dissipativity enforcement via perturbation of para-Hermitian pencils. IEEE Trans. Circuit. Syst. I Regular Paper 60, 164–177 (2012)

    Article  Google Scholar 

  18. Cheng, S.H., Higham, N.J.: The nearest definite pair for the Hermitian generalized eigenvalue problem. Linear Algebra Appl. 302/303, 63–76 (1999). Special issue dedicated to Hans Schneider (Madison, 1998)

    Google Scholar 

  19. Coelho, C.P., Phillips, J.R., Silveira, L.M.: Robust rational function approximation algorithm for model generation. In: Proceedings of the 36th ACM/IEEE Design Automation Conference (DAC), New Orleans, pp. 207–212 (1999)

    Google Scholar 

  20. Crawford, C.R.: The numerical solution of the generalised eigenvalue problem. Ph.D. thesis, University of Michigan, Ann Arbor (1970)

    Google Scholar 

  21. Crawford, C.R.: Algorithm 646: PDFIND: a routine to find a positive definite linear combination of two real symmetric matrices. ACM Trans. Math. Softw. 12, 278–282 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Crawford, C.R., Moon, Y.S.: Finding a positive definite linear combination of two Hermitian matrices. Linear Algebra Appl. 51, pp. 37–48 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  23. Faßbender, H., Mackey, D.S., Mackey, N., Xu, H.: Hamiltonian square roots of skew-Hamiltonian matrices. Linear Algebra Appl. 287, 125–159 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Freiling, G., Mehrmann, V., Xu, H.: Existence, uniqueness and parametrization of Lagrangian invariant subspaces. SIAM J. Matrix Anal. Appl. 23, 1045–1069 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Freund, R.W., Jarre, F.: An extension of the positive real lemma to descriptor systems. Optim. Methods Softw. 19, 69–87 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Green, M., Limebeer, D.J.N.: Linear Robust Control. Prentice-Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  27. Grivet-Talocia, S.: Passivity enforcement via perturbation of Hamiltonian matrices. IEEE Trans. Circuits Syst. 51, pp. 1755–1769 (2004)

    Article  MathSciNet  Google Scholar 

  28. Guglielmi, N., Kressner, D., Lubich, C.: Low rank differential equations for Hamiltonian matrix nearness problems. Oberwolfach preprints, OWP 2013-01 (2013)

    Google Scholar 

  29. Guo, C.-H., Higham, N.J., Tisseur, F.: An improved arc algorithm for detecting definite Hermitian pairs. SIAM J. Matrix Anal. Appl. 31, 1131–1151 (2009)

    Article  MathSciNet  Google Scholar 

  30. Higham, N.J., Tisseur, F., Van Dooren, P.: Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems. Linear Algebra Appl. 351/352, 455–474 (2002). Fourth special issue on linear systems and control

    Google Scholar 

  31. Higham, N.J., Konstantinov, M.M., Mehrmann, V., Petkov, P.Hr.: The sensitivity of computational control problems. Control Syst. Mag. 24, 28–43 (2004)

    Google Scholar 

  32. Karow, M.: μ-values and spectral value sets for linear perturbation classes defined by a scalar product. SIAM J. Matrix Anal. Appl. 32, 845–865 (2011)

    Google Scholar 

  33. Karow, M., Kressner, D., Tisseur, F.: Structured eigenvalue condition numbers. SIAM J. Matrix Anal. Appl. 28, 1052–1068 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Konstantinov, M.M., Mehrmann, V., Petkov, P.Hr.: Perturbation analysis for the Hamiltonian Schur form. SIAM J. Matrix Anal. Appl. 23, 387–424 (2002)

    Google Scholar 

  35. Konstantinov, M.M., Mehrmann, V., Petkov, P.Hr.: Perturbed spectra of defective matrices. J. Appl. Math. 3, pp. 115–140 (2003)

    Google Scholar 

  36. Kressner, D., Peláez, M.J., Moro, J.: Structured Hölder condition numbers for multiple eigenvalues. SIAM J. Matrix Anal. Appl. 31, 175–201 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lancaster, P.: Strongly stable gyroscopic systems. Electron. J. Linear Algebra 5, 53–67 (1999)

    MATH  MathSciNet  Google Scholar 

  38. Lancaster, P., Rodman, L.: The Algebraic Riccati Equation. Oxford University Press, Oxford (1995)

    Google Scholar 

  39. Li, C.-K., Mathias, R.: Distances from a Hermitian pair to diagonalizable and nondiagonalizable Hermitian pairs. SIAM J. Matrix Anal. Appl. 28, 301–305 (2006) (electronic)

    Google Scholar 

  40. Lin, W.-W., Mehrmann, V., Xu, H.: Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl. 301–303, 469–533 (1999)

    Article  MathSciNet  Google Scholar 

  41. Mehl, C., Mehrmann, V., Ran, A., Rodman, L.: Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices. Linear Multilinear Algebra 57, 141–185 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Mehl, C., Mehrmann, V., Ran, A.C.M., Rodman, L.: Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations. Linear Algebra Appl. 435, 687–716 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  43. Mehl, C., Mehrmann, V., Ran, A.C.M., Rodman, L.: Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations. Linear Algebra Appl. 436, 4027–4042 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Mehl, C., Mehrmann, V., Ran, A.C.M., Rodman, L.: Jordan forms of real and complex matrices under rank one perturbations. Oper. Matrices 7, 381–398 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  45. Mehl, C., Mehrmann, V., Ran, A.C.M., Rodman, L.: Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations. BIT 54, 219–255 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  46. Mehrmann, V.: The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Lecture Notes in Control and Information Science, vol. 163. Springer, Heidelberg (1991)

    Google Scholar 

  47. Mehrmann, V.: A step towards a unified treatment of continuous and discrete time control problems. Linear Algebra Appl. 241–243, 749–779 (1996)

    Article  MathSciNet  Google Scholar 

  48. Mehrmann, V., Xu, H.: Perturbation of purely imaginary eigenvalues of Hamiltonian matrices under structured perturbations. Electron. J. Linear Algebra 17, 234–257 (2008)

    MATH  MathSciNet  Google Scholar 

  49. Overton, M., Van Dooren, P.: On computing the complex passivity radius. In: Proceedings of the 4th IEEE Conference on Decision and Control, Seville, pp. 760–7964 (2005)

    Google Scholar 

  50. Paige, C., Van Loan, C.: A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 14, 11–32 (1981)

    Article  Google Scholar 

  51. Pontryagin, L.S., Boltyanskii, V., Gamkrelidze, R., Mishenko, E.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962)

    MATH  Google Scholar 

  52. Ran, A.C.M., Rodman, L.: Stability of invariant maximal semidefinite subspaces. Linear Algebra Appl. 62, 51–86 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  53. Ran, A.C.M., Rodman, L.: Stability of invariant Lagrangian subspaces I. In: Gohberg, I. (ed.) Operator Theory: Advances and Applications, vol. 32, pp. 181–218. Birkhäuser, Basel (1988)

    Chapter  Google Scholar 

  54. Ran, A.C.M., Rodman, L.: Stability of invariant Lagrangian subspaces II. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds.) Operator Theory: Advances and Applications, vol. 40, pp. 391–425. Birkhäuser, Basel (1989)

    Google Scholar 

  55. Rump, S.M.: Eigenvalues, pseudospectrum and structured perturbations. Linear Algebra Appl. 413, 567–593 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  56. Srivastava, R.: Distance problems for Hermitian matrix polynomials – an ε-pseudospectra based approach. PhD thesis, Department of Mathematics, IIT Guwahati (2012)

    Google Scholar 

  57. Stewart, G.W.: Perturbation bounds for the definite generalized eigenvalue problem. Linear Algebra Appl. 23, 69–85 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  58. Stewart, G.W., Sun, J.-G.: Matrix perturbation theory. Computer Science and Scientific Computing. Academic, Boston (1990)

    MATH  Google Scholar 

  59. Sun, J.-G.: A note on Stewart’s theorem for definite matrix pairs. Linear Algebra Appl. 48, 331–339 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  60. Sun, J.-G.: The perturbation bounds for eigenspaces of a definite matrix-pair. Numer. Math. 41, 321–343 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  61. Taussky, O.: Positive-definite matrices. In: Inequalities (Proceeding Symposium Wright-Patterson Air Force Base, Ohio, 1965), pp. 309–319. Academic, New York (1967)

    Google Scholar 

  62. Thompson, R.C.: Pencils of complex and real symmetric and skew matrices. Linear Algebra Appl. 147, pp. 323–371 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  63. Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  64. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    Google Scholar 

  65. Uhlig, F.: On computing the generalized Crawford number of a matrix. Linear Algebra Appl. 438, 1923–1935 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  66. Van Dooren, P.: The generalized eigenstructure problem in linear system theory. IEEE Trans. Autom. Control AC-26, 111–129 (1981)

    Article  Google Scholar 

  67. Van Dooren, P.: A generalized eigenvalue approach for solving Riccati equations. SIAM J. Sci. Stat. Comput. 2, 121–135 (1981)

    Article  MATH  Google Scholar 

  68. Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, Assam, India

    Shreemayee Bora

  2. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Michael Karow

Authors
  1. Shreemayee Bora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Karow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shreemayee Bora .

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

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bora, S., Karow, M. (2015). Structured Eigenvalue Perturbation Theory. 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_8

Download citation

Publish with us

Policies and ethics

Search

Navigation