Advertisement

Ukrainian Mathematical Journal

, Volume 67, Issue 3, pp 464–486 | Cite as

Necessary and Sufficient Conditions for the Existence of Weighted Singular-Valued Decompositions of Matrices with Singular Weights

  • I. V. Sergienko
  • E. F. Galba
  • V. S. Deineka
Article

A weighted singular-valued decomposition of matrices with singular weights is obtained by using orthogonal matrices. The necessary and sufficient conditions for the existence of the constructed weighted singular-valued decomposition are established. The indicated singular-valued decomposition of matrices is used to obtain a decomposition of their weighted pseudoinverse matrices and decompose them into matrix power series and products. The applications of these decompositions are discussed.

Keywords

Linear Algebraic Equation Matrix Power Singular Weight Weighted Pseudoinverse Weighted Normal Pseudosolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. E. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs (1967).zbMATHGoogle Scholar
  2. 2.
    C. F. van Loan, “Generalizing the singular value decomposition,” SIAM J. Numer. Anal., 13, No. 1, 76−83 (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    E. F. Galba, “Weighted singular decomposition and weighted pseudoinversion of matrices,” Ukr. Mat. Zh., 48, No. 10, 1426−1430 (1996); English translation: Ukr. Math. J., 48, No. 10, 1618−1622 (1996).Google Scholar
  4. 4.
    C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs (1974).zbMATHGoogle Scholar
  5. 5.
    E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Weighted singular decomposition and weighted pseudoinversion of matrices with singular weights,” Zh. Vychisl. Mat. Mat. Fiz., 52, No. 12, 2115−2132 (2012).zbMATHGoogle Scholar
  6. 6.
    E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Necessary and sufficient conditions for the existence of one version of weighted singular decomposition of matrices with singular weights,” Dokl. Ros. Akad. Nauk, 455, No. 3, 261−264 (2014).MathSciNetGoogle Scholar
  7. 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).zbMATHMathSciNetGoogle Scholar
  8. 8.
    E. H. Moore, “On the reciprocal of the general algebraic matrix,” Abstr. Bull. Amer. Math. Soc., 26, 394−395 (1920).Google Scholar
  9. 9.
    R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Phil. Soc., 51, No. 3, 406−413 (1955).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    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).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights,” Ukr. Mat. Zh., 63, No. 1, 80−101 (2011); English translation: Ukr. Math. J., 63, No. 1, 125−133 (2011).Google Scholar
  12. 12.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Theorems on existence and uniqueness in the theory of weighted pseudoinversion with singular weights,” Kibernet. Sist. Anal., No. 1, 14−33 (2011).Google Scholar
  13. 13.
    A. Albert, Regression, Pseudoinversion, and Recursive Estimation, Academic Press, New York (1972).Google Scholar
  14. 14.
    E. F. Galba, I. N. Molchanov, and V. V. Skopetskii, “Iterative methods for the evaluation of a weighted pseudoinverse matrix with singular weights,” Kibernet. Sist. Anal., No. 5, 150–169 (1999).Google Scholar
  15. 15.
    E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Limit representations of weighted pseudoinverse matrices with singular weights and the regularization of problems,” Zh. Vychisl. Mat. Mat. Fiz., 44, No. 11, 1928−1946 (2004).zbMATHMathSciNetGoogle Scholar
  16. 16.
    E. F. Galba, “Iterative methods for the determination of weighted normal pseudosolutions with singular weights,” Zh. Vychisl. Mat. Mat. Fiz., 39, No. 6, 882–896 (1999).MathSciNetGoogle Scholar
  17. 17.
    P. Lancaster and P. Rozsa, “Eigenvectors of H-self-adjoint matrices,” Z. Angew. Math. Mech., 64, No. 9, 439−441 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kh. D. Ikramov, “On the algebraic properties of classes of pseudocommutative and H-self-adjoint matrices,” Zh. Vychisl. Mat. Mat. Fiz., 32, No. 8, 155–169 (1992).MathSciNetGoogle Scholar
  19. 19.
    Kh. D. Ikramov, Problems of Linear Algebra with Generalized Symmetries and Numerical Algorithms for Their Solution [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Moscow (1991).Google Scholar
  20. 20.
    P. Lancaster, Theory of Matrices, Academic Press, New York (1969).zbMATHGoogle Scholar
  21. 21.
    E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Decompositions and polynomial limit representations of weighted pseudoinverse matrices,” Zh. Vychisl. Mat. Mat. Fiz., 47, No. 5, 747−766 (2007).zbMATHMathSciNetGoogle Scholar
  22. 22.
    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (1986).Google Scholar
  23. 23.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Decomposition of weighted pseudoinverse matrices in matrix power products,” Ukr. Mat. Zh., 56, No. 11, 1539−1556 (2004); English translation: Ukr. Math. J., 56, No. 11, 1828−1848 (2004).Google Scholar
  24. 24.
    A. N. Tikhonov and V. Ya. Arsenin, Methods for the Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1986).Google Scholar
  25. 25.
    A. I. Zhdanov, “Method of extended regularized normal equations,” Zh. Vychisl. Mat. Mat. Fiz., 52, No. 2, 205−208 (2012).zbMATHMathSciNetGoogle Scholar
  26. 26.
    V. A. Morozov, Regular Methods for the Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1987).Google Scholar
  27. 27.
    G. M. Vainikko and A. Yu. Veretennikov, Iterative Procedures in Ill-Posed Problems [in Russian], Nauka, Moscow (1986).Google Scholar
  28. 28.
    E. V. Arkharov and R. A. Shafiev, “Methods of regularization of the problem of connected pseudoinversion with approximate data,” Zh. Vychisl. Mat. Mat. Fiz., 43, No. 3, 347–353 (2003).zbMATHMathSciNetGoogle Scholar
  29. 29.
    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).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • I. V. Sergienko
    • 1
  • E. F. Galba
    • 1
  • V. S. Deineka
    • 1
  1. 1.Institute of CyberneticsUkrainian National Academy of SciencesKievUkraine

Personalised recommendations