Skip to main content
SpringerLink
Log in

Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights

  • Published:
Ukrainian Mathematical Journal Aims and scope

For one definition of weighted pseudoinversion with singular weights, we establish necessary and sufficient conditions for the existence and uniqueness of a solution of a system of matrix equations. Expansions of weighted pseudoinverse matrices in matrix power series and matrix power products are obtained. A relationship between weighted pseudoinverse matrices the weighted normal pseudosolutions is established. Iterative methods for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions are constructed.

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. F. Ward, T. L. Boullion, and T. O. Lewis, “Weighted pseudoinverses with singular weights,” SIAM J. Appl. Math., 21, No. 3, 480–482 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  2. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Representations and expansions of weighted pseudoinverse matrices, iterative methods, and regularization of problems. II. Singular weights,” Kibernet. Sist. Analiz, 44, No. 3, 75–102 (2008).

    MathSciNet  Google Scholar 

  3. E. F. Galba, “Weighted pseudoinversion of matrices with singular weights,” Ukr. Mat. Zh., 46, No. 10, 1323–1327 (1994); English translation: Ukr. Math. J., 46, No. 10, 1457–1462 (1994).

    Article  MathSciNet  Google Scholar 

  4. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Limit representations of weighted pseudoinverse matrices with singular weights and regularization of problems,” Zh. Vychisl. Mat. Mat. Fiz., 44, No. 11, 1928–1946 (2004).

    MathSciNet  MATH  Google Scholar 

  5. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods,” Ukr. Mat. Zh. , 59, No. 9, 1269–1290 (2007); English translation: Ukr. Math. J., 59, No. 9, 1417–1440 (2007).

    Article  MathSciNet  Google Scholar 

  6. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Expansions and polynomial limit representations of weighted pseudoinverse matrices,” Zh. Vychisl. Mat. Mat. Fiz., 47, No. 5, 747–766 (2007).

    MathSciNet  Google Scholar 

  7. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights,” Zh. Vychisl. Mat. Mat. Fiz., 49, No. 8, 1347–1363 (2009).

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc., 51, No. 3, 406–413 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Albert, Regression, Pseudoinversion, and Recursive Estimation [Russian translation], Nauka, Moscow (1977).

    Google Scholar 

  11. E. F. Galba, I. N. Molchanov, and V. V. Skopetskii, “Iterative methods for computing a weighted pseudoinverse matrix with singular weights,” Kibernet. Sist. Analiz, 35, No. 5, 150–169 (1999).

    MathSciNet  Google Scholar 

  12. E. F. Galba, “Iterative methods for computing a weighted normal pseudosolution with singular weights,” Zh. Vychisl. Mat. Mat. Fiz., 39, No. 6, 882–896 (1999).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Kh. D. Ikramov, “On algebraic properties of classes of pseudopermutation and H-self-adjoint matrices,” Zh. Vychisl. Mat. Mat. Fiz., 32, No. 8, 1155–1169 (1992).

    MathSciNet  MATH  Google Scholar 

  15. F. R. Gantmakher, The Theory of Matrices [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  16. H. P. Decell, “An application of the Cayley–Hamilton theorem to generalized matrix inversion,” SIAM Rev., 7, No. 4, 526–528 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (1985).

    MATH  Google Scholar 

  18. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Fast converging iterative methods for computing weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights,” Zh. Vychisl. Mat. Mat. Fiz., 45, No. 10, 1731–1755 (2005).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Cybernetics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    I. V. Sergienko, E. F. Galba & V. S. Deineka

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

  3. V. S. Deineka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 1, pp. 80–101, January, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sergienko, I.V., Galba, E.F. & Deineka, V.S. Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights. Ukr Math J 63, 98–124 (2011). https://doi.org/10.1007/s11253-011-0490-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0490-3

Keywords

Search

Navigation