Skip to main content

The condition number of equivalence transformations that block diagonalize matrix pencils

Section A.1 Of General (A-λB)-Pencils

Part of the Lecture Notes in Mathematics book series (LNM,volume 973)

Abstract

How ill-conditioned must a matrix S be if its columns are constrained to span certain subspaces? We answer this question in order to find nearly best conditioned matrices S R and S L that block diagonalize a given matrix pencil T=AB, i.e. S −1L TS R=Θ is bloc diagonal. We show that the best conditioned S R has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of Θ. Thus, the more nearly the right subspaces overlap the more ill-conditioned S R must be. The same is true of S L and the left subspaces. For the standard eigenproblem (T=A−λI), S L = S R and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number, namely the norm of the projection matrix associated with one of the subspaces. We apply this result to bound the error in an algorithm to compute analytic functions of matrices, for instance exp(T).

Keywords

  • Condition Number
  • Invariant Subspace
  • Positive Definite Matrix
  • Diagonal Block
  • Perturbation Bound

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. BART, I. GOHBERG, M. A. KAASHOEK, P. VAN DOOREN, Factorizations of Transfer Functions, SIAM J. Control, vol 18, no 6, November 1980, pp 675–696

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. F. L. BAUER, A further generalization of the Kantorovic inequality, Numer. Math., 3, pp 117–119, 1961

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. F. L. BAUER, Optimally scaled matrices, Numer. Math., 5, pp. 73–87, 1983

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. F. L. BAUER, C. T. FIKE, Norms and Exclusion Theorems, Numer. Math., 2, pp 137–141, 1960

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. F. L. BAUER, A. S. HOUSEHOLDER, Some inequalities involving the euclidean condition of a matrix, Numer. Math., 2, pp 308–311, 1960

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. C. DAVIS, W. KAHAN, Some new bounds on perturbations of subspaces, Bull. A. M. S., 75, 1969, pp 863–8

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J. DEMMEL, The Condition Number of Similarities that Diagonalize Matrices, Electronics Research Laboratory Memorandum, University of California, Berkeley, 1982

    Google Scholar 

  8. F. R. GANTMACHER, The Theory of Matrices, trans. K. A. Hirsch, Chelsea, 1959, vol. 2

    Google Scholar 

  9. E. ISAACSON, H. B. KELLER, Analysis of Numerical Methods, Wiley, 1966

    Google Scholar 

  10. W. KAHAN, Conserving Confluence Curbs Ill-Condition, Technical Report 6, Computer Science Dept., University of California, Berkeley, August 4, 1972

    Google Scholar 

  11. T. KATO, Estimation of Iterated Matrices, with Application to the von Neumann Condition, Numer. Math., 2, pp 22–29, 1980

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, 1966

    Google Scholar 

  13. A. RUHE, Properties of a Matrix with a Very Ill-conditioned Eigenproblem, Numer. Math., 15, pp 57–60, 1970

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. A. VAN DER SLUIS, Condition Numbers and Equilibration of Matrices, Numer. Math., 14, pp 14–23, 1969

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. R. A. SMITH, The Condition Numbers of the Matrix Eigenvalue Problem, Numer. Math., 10, pp 232–240, 1967

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. G. W. STEWART, Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems, SIAM Review, vol. 15, no. 4, Oct 1973, p 752

    CrossRef  MathSciNet  Google Scholar 

  17. J. G. SUN, The Perturbation Bounds for Eigenspaces of Definite Matrix Pairs, to appear

    Google Scholar 

  18. P. VAN DOOREN, P. DEWILDE, Minimal Cascade Factorization of Real and Complex Rational Transfer Matrices, IEEE Trans. on Circuits and Systems, vol. CAS-28, no. 5, May 1981, p 395.

    MathSciNet  MATH  Google Scholar 

  19. J. H. WILKINSON, Rounding Errors in Algebraic Processes, Prentice Hall, 1963

    Google Scholar 

  20. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, 1965

    Google Scholar 

  21. J. H. WILKINSON, Note on Matrices with a Very Ill-Conditioned Eigenproblem, Numer. Math., 19, pp 176–178, 1972

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Demmel, J. (1983). The condition number of equivalence transformations that block diagonalize matrix pencils. In: Kågström, B., Ruhe, A. (eds) Matrix Pencils. Lecture Notes in Mathematics, vol 973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062091

Download citation

  • DOI: https://doi.org/10.1007/BFb0062091

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11983-8

  • Online ISBN: 978-3-540-39447-1

  • eBook Packages: Springer Book Archive