Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 185–192 | Cite as

Representing Weighted Pseudoinverse Matrices with Mixed Weights in Terms of Other Pseudoinverses

SYSTEMS ANALYSIS
  • 3 Downloads

Abstract

The paper considers weighted pseudoinverse where both weighted matrices are symmetric and one of them is positive definite matrix and the other is nonsingular and indefinite. Formulas are obtained to represent these matrices in terms of the Moore–Penrose pseudoinverse matrix and other weighted pseudoinverses.

Keywords

weighted pseudoinverses with sign-indefinite weights Moore–Penrose pseudoinverses weighted pseudoinverses with mixed weights 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. N. Khimich, E. F. Galba, and N. A. Varenyuk, “Weighted pseudoinverses with sign-indefinite weights,” Dopov. Nac. Akad. Nauk Ukr., No. 6, 14–20 (2017).Google Scholar
  2. 2.
    E. H. Moore, “On the reciprocal of the general algebraic matrix,” Abstract Bull. Amer. Math. Soc., Vol. 26, 394–395 (1920).Google Scholar
  3. 3.
    R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Phil. Soc., Vol. 51, No. 3, 406–413 (1955).CrossRefMATHGoogle Scholar
  4. 4.
    J. F. Ward, T. L. Boullion, and T. O. Lewis, “Weighted pseudoinverses with singular weights,” SIAM J. Appl. Math., Vol. 21, No. 3, 480–482 (1971).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. I. Positive definite weights,” Cybern. Syst. Analysis, Vol. 44, No. 1, 36–55 (2008).CrossRefMATHGoogle Scholar
  6. 6.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights,” Cybern. Syst. Analysis, Vol. 44, No. 3, 375–397 (2008).CrossRefMATHGoogle Scholar
  7. 7.
    A. Albert, Regression, Pseudoinversion, and Recurrent Estimation [in Russian], Nauka, Moscow (1977).Google Scholar
  8. 8.
    M. Z. Nashed, Generalized Inverses and Applications, Academic Press, New York (1976).MATHGoogle Scholar
  9. 9.
    A. Ben-Izrael and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer Verlag, New York (2003).MATHGoogle Scholar
  10. 10.
    J. Wilkinson and K. Reinsch, A Handbook of Algol Algorithms. Linear Algebra [Russian translation], Mashinostroenie, Moscow (1976).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations