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Bounds for the Distance Between Nearby Jordan and Kronecker Structures in a Closure Hierarchy

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Abstract

Computing the fine-canonical-structure elements of matrices and matrix pencils are ill-posed problems. Therefore, besides knowing the canonical structure of a matrix or a matrix pencil, it is equally important to know what are the nearby canonical structures that explain the behavior under small perturbations. Qualitative strata information is provided by our StratiGraph tool. Here, we present lower and upper bounds for the distance between Jordan and Kronecker structures in a closure hierarchy of an orbit or bundle stratification. This quantitative information is of importance in applications, e.g., distance to more degenerate systems (uncontrollability). Our upper bounds are based on staircase regularizing perturbations. The lower bounds are of Eckart―Young type and are derived from a matrix representation of the tangent space of the orbit of a matrix or a matrix pencil. Computational results illustrate the use of the bounds. Bibliography: 42 titles.

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REFERENCES

  1. S. Abeasis and A. Del Fra, “Degenerations for the representations of a quiver of type Am," J. Algebra, 93, 376–412 (1985).

    Google Scholar 

  2. V. I. Arnold, “On matrices depending on parameters," Russian Math. Surveys, 26, 29–43 (1971).

    Google Scholar 

  3. T. Beelen and P. Van Dooren, “An improved algorithm for the computation of Kronecker's canonical form of a singular pencil," Linear Algebra Appl., 105, 9–65 (1988).

    Google Scholar 

  4. V. A. Belyi, V. B. Khazanov, and V. N. Kublanovskaya, “Spectral problems for matrix pencils. Methods and algorithms. III," Sov. J. Numer. Anal. Math. Modelling, 4, 19–51 (1989).

    Google Scholar 

  5. D. Boley, “Estimating the sensitivity of the algebraic structure of pencils with simple eigenvalue estimates," SIAM J. Matrix Anal. Appl., 11(4), 632–643 (1990).

    Google Scholar 

  6. D. Boley, “The algebraic structure of pencils and block Toeplitz matrices, Linear Algebra Appl., 279, 255–279 (1998).

    Google Scholar 

  7. K. Bongartz, “On degenerations and extensions of finite dimensional modules," Adv. Math., 121(2), 245–287 (1996).

    Google Scholar 

  8. R. Byers, C. He, and V. Mehrmann, “Where is the nearest non-regular pencil?" Linear Algebra Appl., 285, 81–105 (1998).

    Google Scholar 

  9. I. De Hoyos, “Points of continuity of the Kronecker canonical form," SIAM J. Matrix Anal. Appl., 11(2), 278–300 (1990).

    Google Scholar 

  10. J. Demmel and A. Edelman, “The dimension of matrices (matrix pencils) with given Jordan (Kronecker) canonical forms," Linear Algebra Appl., 230, 61–87 (1995).

    Google Scholar 

  11. J. Demmel and B. Kågström," “The generalized Schur decomposition of an arbitrary pencil A-?B: robust software with error bounds and applications. Part I: Theory and algorithms," ACM Trans. Math. Software, 19, No. 2, 160–174 (1993).

    Google Scholar 

  12. J. Demmel and B. Kågström, “The generalized Schur decomposition of an arbitrary pencil A-?B: robust software with error bounds and applications. Part II: Software and applications," ACM Trans. Math. Software, 19, No. 2, 175–201 (1993).

    Google Scholar 

  13. A. Edelman, T. Arias, and S. Smith, “The geometry of algorithms with orthogonality constraints," SIAM J. Matrix Anal. Appl., 20(2), 303–353 (1999).

    Google Scholar 

  14. A. Edelman, E. Elmroth, and B. Kågström, “A geometric approach to perturbation theory of matrices and matrix pencils. Part I: Versal deformations," SIAM J. Matrix Anal. Appl., 18(3), 653–692 (1997).

    Google Scholar 

  15. A. Edelman, E. Elmroth, and B. Kågström, “A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm," SIAM J. Matrix Anal. Appl., 20(3), 667–699 (1999).

    Google Scholar 

  16. E. Elmroth, “On the stratification of the Kronecker canonical form," Report UMINF-95.14, Department of Computing Science, Umeå University, S-901 87, Umeå, Sweden (1995).

  17. E. Elmroth, P. Johansson, and B. Kågström, “Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures," Numer. Linear Algebra Appl., accepted (2000).

  18. E. Elmroth and B. Kågström, “The set of 2-by-3 matrix pencils-Kronecker structures and their transitions under perturbations," SIAM J. Matrix Anal. Appl., 17(1), 1–34 (1996).

    Google Scholar 

  19. F. Gantmacher, The Theory of Matrices, Vols. I and II [Engl. translation] Chelsea, New York (1959).

    Google Scholar 

  20. M. I. Garcia-Planas and M. D. Margret, “An alternative system of structural invariants of quadruples of matrices," Linear Algebra Appl., 291, 83–102 (1999).

    Google Scholar 

  21. J. M. Gracia and I. De Hoyos, “Safety neighbourhoods for the Kronecker canonical form," SIAM J. Matrix Anal. Appl., submitted (1999).

  22. J. M. Gracia, I. De Hoyos, and F. E. Velasco, “Safety neighbourhoods for the invariants of the matrix similarity," to appear in Linear Multilinear Algebra (1996).

  23. J. M. Gracia, I. De Hoyos, and Z. Zaballa, “Perturbation of linear control systems," Linear Algebra Appl., 121, 353–383 (1989).

    Google Scholar 

  24. D. Hinrichsen and J. O. Halloran, “Orbit closures of singular matrix pencils," J. Pure Appl. Algebra, 81, 117–137 (1992).

    Google Scholar 

  25. P. Johansson, StratiGraph User's Guide. Version 1.1, Report UMINF-99.11, Department of Computing Science, Umeå University, SE-901 87, Umeå, Sweden (2000).

  26. B. Kågström, “RGSVD-an algorithm for computing the Kronecker canonical form and reducing subspaces of singular matrix pencils A-?B," SIAM J. Sci. Stat. Comp., 7(1), 185–211 (1986).

    Google Scholar 

  27. B. Kågström, “Singular matrix pencils," in: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai et al., eds., to appear in SIAM Publications, Philadelphia (2000).

  28. B. Kågström and A. Ruhe, “ALGORITHM 560: An algorithm for the numerical computation of the Jordan normal form of a complex matrix [F2]," ACM Trans. Math. Software, 6(3), 437–443 (1980).

    Google Scholar 

  29. B. Kågström and A. Ruhe, “An algorithm for the numerical computation of the Jordan normal form of a complex matrix," ACM Trans. Math. Software, 6(3), 389–419 (1980).

    Google Scholar 

  30. B. Kågström and P. Wiberg, “Extracting partial canonical structure for large scale eigenvalue problems," Numer. Algorithms, 24(3), 195–237 (2000).

    Google Scholar 

  31. V. B. Khazanov and V. N. Kublanovskaya, “Spectral problems for matrix pencils. Methods and algorithms I," Sov. J. Numer. Anal. Math. Modelling, 3, 337–371 (1988).

    Google Scholar 

  32. V. B. Khazanov and V. N. Kublanovskaya, “Spectral problems for matrix pencils. Methods and algorithms II," Sov. J. Numer. Anal. Math. Modelling, 3, 467–484 (1988).

    Google Scholar 

  33. V. N. Kublanovskaya, “On a method of solving the complete eigenvalue problem for a degenerate matrix," Zh. Vychisl. Mat. Mat. Fiz., 6, 611–620 (1966).

    Google Scholar 

  34. V. N. Kublanovskaya, “An approach to solving the spectral problem of A - ?B," in: Lect. Notes. Math., Vol. 973, Springer, Berlin (1983), pp. 17–29.

    Google Scholar 

  35. V. N. Kublanovskaya, “AB-algorithm and its modifications for the spectral problem of linear pencils of ma-trices," Numer. Math., 43, 329–342 (1984).

    Google Scholar 

  36. V. N. Kublanovskaya, “Rank-division algorithms and their applications," J. Numer. Linear Algebra Appl., 1(2), 199–213 (1992).

    Google Scholar 

  37. V. N. Kublanovskaya, “An approach to solving multiparameter algebraic problems," J. Math. Sci., 89(6), 1715–1749 (1998).

    Google Scholar 

  38. R. Lippert and A. Edelman, “Nonlinear eigenvalue problems," to appear in: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai et al., eds., SIAM Publications, Philadelphia (2000).

    Google Scholar 

  39. A. Pokrzywa, “On perturbations and the equivalence orbit of a matrix pencil," Linear Algebra Appl., 82, 99–121 (1986).

    Google Scholar 

  40. A. Ruhe, “An algorithm for numerical determination of the structure of a general matrix," BIT, 10, 196–216 (1970).

    Google Scholar 

  41. P. Van Dooren, “The computation of Kronecker's canonical form of a singular pencil," Linear Algebra Appl., 27, 103–141 (1979).

    Google Scholar 

  42. J. C. Willems, “Topological classification and structural stability of linear systems," J. Diff. Equations, 35, 306–318 (1980).

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing Science, Umeå University, SE-901 87, Umeå, Sweden

    E. Elmroth, P. Johansson & B. Kågström

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  1. E. Elmroth
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  2. P. Johansson
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  3. B. Kågström
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Elmroth, E., Johansson, P. & Kågström, B. Bounds for the Distance Between Nearby Jordan and Kronecker Structures in a Closure Hierarchy. Journal of Mathematical Sciences 114, 1765–1779 (2003). https://doi.org/10.1023/A:1022498301583

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