Skip to main content
Log in

Weighted pseudoinverses and weighted normal pseudosolutions with singular weights

Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J. F. Ward, T. L. Boullion, and T. O. Lewis, “Weighted Pseudoinverses with Singular Weights,” SIAM J. Appl. Math. 21(3), 480–482 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  2. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Expansions and Polynomial Limit Representations of Weighted Pseudoinverses,” Zh. Vychisl. Mat. Mat. Fiz. 47, 747–766 (2007) [Comput. Math. Math. Phys. 47, 713–731 (2007)].

    MathSciNet  Google Scholar 

  3. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Representations and Expansions of Weighted Pseudoinverse Matrices, Iterative Methods, and Problem Regularization: 2. Singular Weights,” Kibern. Sistemn. Anal., No. 3, 75–102 (2008).

  4. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Representations and Expansions of Weighted Pseudoinverse Matrices, Iterative Methods, and Problem Regularization: 1. Positive Definite Weights,” Kibern. Sistemn. Anal., No. 1, 47–73 (2008).

  5. J. S. Chipman, “On Least Squares with Insufficient Observation,” J. Amer. Statist. Assoc. 59(308), 1078–1111 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. H. Moore, “On the Reciprocal of the General Algebraic Matrix,” Abstract. Bull. Amer. Math. Soc. 26, 394–395 (1920).

    Google Scholar 

  7. R. Penrose, “A Generalized Inverse for Matrices,” Proc. Phil. Soc., Cambridge 51, 406–413 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. E. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic Press, New York, 1972; Nauka, Moscow, 1975).

    MATH  Google Scholar 

  9. E. F. Galba, I. N. Molchanov, and V. V. Skopetskii, “Iterative Methods for Calculating a Weighted Pseudoinverse with Singular Weights,” Kibern. Sistemn. Anal., No. 5, 150–169 (1999).

    Google Scholar 

  10. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Limit Representations of Weighted Pseudoinverses with Singular Weights and the Regularization of Problems,” Zh. Vychisl. Mat. Mat. Fiz. 44, 1928–1946 (2004) [Comput. Math. Math. Phys. 44, 1833–1850 (2004)].

    MATH  MathSciNet  Google Scholar 

  11. E. F. Galba, “Iterative Methods for Computing Weighted Minimum-Length Least Squares Solution with a Singular Weight Matrix,” Zh. Vychisl. Mat. Mat. Fiz. 39, 882–896 (1999) [Comput. Math. Math. Phys. 39, 848–861 (1999)].

    MathSciNet  Google Scholar 

  12. P. Lancaster and P. Rozsa, “Eigenvectors of H-Self-Adjoint Matrices,” Z. angew. Math. Mech. 64, 439–441 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  13. Kh. D. Ikramov, “On the Algebraic Properties of the Classes of Pseudocommuting and H-Self-Adjoint Matrices,” Zh. Vychisl. Mat. Mat. Fiz. 32(8), 155–169 (1992).

    MathSciNet  Google Scholar 

  14. F. R. Gantmakher, The Theory of Matrices (Chelsea, New York; Nauka, Moscow, 1967, 1959).

    Google Scholar 

  15. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Limit Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights and Regularization of Problems,” Kibern. Sistemn. Anal., No. 6, 46–65 (2003).

  16. H. P. Decell, “An Application of the Cayley-Hamilton Theorem to Generalized Matrix Inversion,” SIAM Rev. 7, 526–528 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  17. A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1974; Winston, Washington, 1977).

    Google Scholar 

  18. L. Elden, “A Weighted Pseudoinverse Generalized Singular Values and Constrained Least Squares Problems,” BIT 22, 487–502 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. A. Morozov, Regular Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  20. E. V. Arkharov and R. A. Shafiev, “Regularization Methods for the Constrained Pseudoinversion Problem with Inaccurate Data,” Zh. Vychisl. Mat. Mat. Fiz. 43, 347–353 (2003) [Comput. Math. Math. Phys. 43, 331–337 (2003)].

    MATH  MathSciNet  Google Scholar 

  21. Kh. D. Ikramov and M. Matin far, “On Computer-Algebra Procedures for Linear Least Squares Problems with Linear Equality Constraints,” Zh. Vychisl. Mat. Mat. Fiz. 44, 206–212 (2004) [Comput. Math. Math. Phys. 44, 190–196 (2004)].

    MATH  MathSciNet  Google Scholar 

  22. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge Univ. Press, Cambridge, 1985; Mir, Moscow, 1989).

    MATH  Google Scholar 

  23. I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products,” Ukr. Mat. Zh. 56, 1539–1556 (2004).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. S. Deineka.

Additional information

Original Russian Text © E.F. Galba, V.S. Deineka, I.V. Sergienko, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 8, pp. 1347–1363.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Galba, E.F., Deineka, V.S. & Sergienko, I.V. Weighted pseudoinverses and weighted normal pseudosolutions with singular weights. Comput. Math. and Math. Phys. 49, 1281–1297 (2009). https://doi.org/10.1134/S0965542509080016

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542509080016

Key words

Navigation