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Completion of a Matrix so that the Inverse has Minimum Norm. Application to the Regularization of Descriptor Control Problems

  • L. Elsner
  • C. He
  • V. Mehrmann
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 62)

Abstract

We discuss the problem of minimizing the spectral norm of the inverse of a matrix, when a submatrix of the matrix can be chosen arbitrarily. In a recent paper by the authors [3], it was shown that the solution of this problem can be discussed in a similar way as the problem of minimizing the norm of the matrix in terms of matrix Riccati inequalities.

Here we review the results for the norm of the inverse and then apply these results to the robust regularization of descriptor control problems. We also describe a numerical method and give numerical examples.

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References

  1. [1]
    A. Bunse-Gerstner, V. Mehrmann, and N. Nichols, Numerical methods for the regularization of descriptor systems by output feedback, Tech. Rep. 987, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455, USA, 1992.Google Scholar
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    A. Bunse-Gerstner, V. Mehrmann, and N. Nichols, Regularization of descriptor systems by derivative and proportional state feedback, SIAM Journal Matrix Analysis and Applications, 13 (1992), pp. 46–67.MathSciNetzbMATHCrossRefGoogle Scholar
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    L. Elsner, C. He, and V. Mehrmann, Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion, Tech. Rep. 92–028, Sonderforschungsbereich 343, Diskrete Strukturen in der Mathematik, Universität Bielefeld, Fakultät für Mathematik.Google Scholar
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    G. Golub AND C.V. Loan, Matrix Computations, The Johns Hopkins Press, Baltimore, Maryland, second ed., 1989.zbMATHGoogle Scholar
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    D. Ouellette, Schur complements and statistics, Linear Algebra and its Applications, 36 (1981), pp. 187–295.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • L. Elsner
    • 1
  • C. He
    • 2
  • V. Mehrmann
    • 2
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Fachbereich MathematikTU Chemnitz-ZwickauChemnitzGermany

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