Skip to main content
SpringerLink
Log in

Weighted Singular Value Decomposition of Matrices with Singular Weights Based on Weighted Orthogonal Transformations

  • SYSTEMS ANALYSIS
  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

Two variants of weighted singular value decompositions of matrices with singular weights using weighted orthogonal matrices are obtained and analyzed. Based on this singular value decompositions of matrices, decomposition of weighted pseudoinverse matrices with singular weights and decomposition of these matrices into matrix power series and products are obtained. The application of these decompositions is determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Forsythe and K. Moler, Computer Solution of Linear Algebraic Systems, Prentice Hall (1967).

  2. C. F. Van Loan, “Generalizing the singular value decomposition,” SIAM J. Numer. Anal., 13, No. 1, 76–83 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  3. E. F. Galba, “Weighted singular decomposition and weighted pseudoinversion of matrices,” Ukr. Math. J., 48, No. 10, 1618–1622 (1996).

    Article  MathSciNet  Google Scholar 

  4. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Classics in Applied Mathematics, SIAM (1987).

  5. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Weighted singular value decomposition and weighted pseudoinversion of matrices with singular weights,” Zh. Vych. Mat. Mat. Fiz., 52, No. 12, 2115–2132 (2012).

    MATH  Google Scholar 

  6. A. N. Khimich, “Perturbation bounds for the least squares problem,” Cybern. Syst. Analysis, 32, No. 3, 434–436 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. N. Khimich and E. A. Nikolaevskaya, “Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data,” Cybern. Syst. Analysis, 44, No. 6, 863–874 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  8. E. A. Nikolaevskaya and A. N. Khimich, “Error estimation for a weighted minimum-norm least squares solution with positive definite weights,” Comp. Math. Math. Physics, 49, No. 3, 409–417 (2009).

    Article  MathSciNet  Google Scholar 

  9. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Necessary and sufficient existence conditions for a weighted singular value decomposition of matrices with singular weights,” Dokl. RAN, 455, No. 3, 261–264 (2014).

    MathSciNet  Google Scholar 

  10. E. H. Moore, “On the reciprocal of the general algebraic matrix,” Abstract. Bull. Amer. Math. Soc., 26, 394–395 (1920).

    Google Scholar 

  11. R. A. Penrose, “Generalized inverse for matrices,” Proc. Cambridge Phil. Soc., 51, No. 3, 406–413 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Albert, Regression, Pseudoinversion, and Recurrent Estimation [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  13. E. F. Galba, I. N. Molchanov, and V. V. Skopetskii, “Methods of computing weighted pseudoinverse matrices with singular weights,” Cybern. Syst. Analysis, 35, No. 5, 814–831 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  14. E. F. Galba, “Iterative methods to evaluate weighted normal pseudosolution with singular weights,” Zh. Vych. Mat. Mat. Fiz., 39, No. 6, 882–896 (1999).

    MathSciNet  Google Scholar 

  15. P. Lancaster and P. Rozsa, “Eigenvectors of H-self-adjoint matrices,” Z. Angew. Math. und Mech., 64, No. 9, 439–441 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  16. Kh. D. Ikramov, “On the algebraic properties of classes of pseudopermutation and Í-self-adjoint matrices,” Zh. Vych. Mat. Mat. Fiz., 32, No. 8, 155–169 (1992).

  17. Kh. D. Ikramov, A Problem Book on Linear Algebra [in Russian], Nauka, Moscow (1975).

  18. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press (1990).

    MATH  Google Scholar 

  19. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Weighted pseudoinverses and weighted normal pseudosolutions with singular weights,” Comp. Math. Math. Physics, 49, No. 8, 1281–1297 (2009).

    Article  MathSciNet  Google Scholar 

  20. I. V. Sergienko, Ye. F. Galba, and V. S. Deineka, “ Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights,” Cybern. Syst. Analysis, 47, No. 1, 11–28 (2011).

  21. A. N. Tikhonov and V. Ya. Arsenin, Methods to Solve Ill-Posed Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  22. V. A. Morozov, Regular Methods to Solve Ill-Posed Problems [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  23. A. I. Zhdanov, “The method of augmented regularized normal equations,” Comp. Math. Math. Physics, 52, No. 2, 194–197 (2012).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    I. V. Sergienko & E. F. Galba

Authors
  1. I. V. Sergienko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. F. Galba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Sergienko.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2015, pp. 28–43.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sergienko, I.V., Galba, E.F. Weighted Singular Value Decomposition of Matrices with Singular Weights Based on Weighted Orthogonal Transformations. Cybern Syst Anal 51, 514–528 (2015). https://doi.org/10.1007/s10559-015-9743-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-015-9743-8

Keywords

Search

Navigation