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Reducing subspaces: Definitions, properties and algorithms

  • Section A.1 Of General (A-λB)-Pencils
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Book cover Matrix Pencils

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

Abstract

In this paper we introduce the new concept of reducing subspaces of a singular pencil, which extends the notion of deflating subspaces to the singular case. We briefly discuss uniqueness of such subspaces and we give an algorithm for computing them. The algorithm also gives the Kronecker canonical form of the singular pencil.

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References

  1. BOLEY D., Computing the controllability/observability decomposition of a linear time invariant dynamic system, a numerical approach, Ph. D. Thesis, Stanford University, 1981.

    Google Scholar 

  2. EMAMI-NAEINI A., VAN DOOREN P., Computation of zeros of linear multivariable systems, to appear Automatica, 1982.

    Google Scholar 

  3. FORNEY, G. D. Jr., Minimal bases of rational vector spaces with applications to multivariable linear systems, SIAM J. Contr., Vol. 13, pp. 493–520, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  4. GANTMACHER F. R., Theory of matrices I & II, Chelsea, New York, 1959.

    MATH  Google Scholar 

  5. MOLER C., STEWART G., An algorithm for the generalized matrix eigenvalue problem, SIAM J. Num. Anal., Vol. 10, pp. 241–256, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  6. PAIGE C., Properties of numerical algorithms related to computing controllability, IEEE Trans. Aut. Contr., Vol. AC-26, pp. 130–138.

    Google Scholar 

  7. STEWART G., Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev., Vol. 15, pp. 727–764, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  8. STEWART G., On the sensitivity of the eigenvalue problem Ax=λBx, SIAM Num. Anal. Vol. 9, pp. 669–686, 1972.

    Article  MathSciNet  Google Scholar 

  9. VAN DOOREN P., The computation of Kronecker's canonical form of a singular pencil, Lin. Alg. & Appl., Vol. 27, pp. 103–141, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  10. VAN DOOREN P., The generalized eigenstructure problem in linear system theory, IEEE Trans. Aut. Contr., Vol. AC-26, pp. 111–129, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  11. VAN DOOREN P., A generalized eigenvalue approach for solving Riccati equations, SIAM Sci. St. Comp., Vol. 2, pp. 121–135, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  12. WILKINSON J., Linear differential equations and Kronecker's canonical form, Recent Advances in Numerical Analysis, Ed. C. de Boor, G. Golub, Academic Press, New York, 1978.

    Google Scholar 

  13. WILKINSON J., Kronecker's canonical form and the QZ algorithm, Lin. Alg. & Appl., Vol. 28, pp. 285–303, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  14. WONHAM W., Linear multivariable theory. A geometric approach, (2nd Ed.) Springer, New York, 1979.

    Book  MATH  Google Scholar 

  15. KUBLANOVSKAYA V., AB algorithm and its modifications for the spectral problem of linear pencils of matrices, LOMI-preprint E-10-81, USSR Academy of Sciences, 1981.

    Google Scholar 

  16. KUBLANOVSKAYA V., On an algorithm for the solution of spectral problems of linear matrix pencils, LOMI-preprint E-1-82, USSR Academy of Sciences, 1982.

    Google Scholar 

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Authors
  1. Paul Van Dooren
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Bo Kågström Axel Ruhe

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© 1983 Springer-Verlag

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Van Dooren, P. (1983). Reducing subspaces: Definitions, properties and algorithms. In: Kågström, B., Ruhe, A. (eds) Matrix Pencils. Lecture Notes in Mathematics, vol 973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062094

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  • DOI: https://doi.org/10.1007/BFb0062094

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11983-8

  • Online ISBN: 978-3-540-39447-1

  • eBook Packages: Springer Book Archive

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