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Existence and Uniqueness of Weighted Pseudoinverses with Nonsingular Indefinite Weights

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Cybernetics and Systems Analysis Aims and scope

Abstract

For arbitrary complex matrices, necessary and sufficient conditions for the existence and uniqueness of weighted pseudoinverses with nonsingular indefinite weights are obtained. The representations of these matrices are obtained in terms of the coefficients of characteristic polynomials of the Hermitizable matrices.

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References

  1. J. S. Chipman, “On least squares with insufficient observation,” J. of the American Statististical Assoc., Vol. 59, No. 308, 1078–1111 (1964).

    Article  MathSciNet  Google Scholar 

  2. 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).

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  4. 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. Math. J., Vol. 63, No. 1, 98–124 (2011).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. I. V. Sergienko and E. F. Galba, “Weighted pseudoinversion with singular weights,” Cybern. Syst. Analysis, Vol. 52, No. 5, 708–729 (2016).

    Article  MathSciNet  Google Scholar 

  7. E. F. Galba and I. V. Sergienko, “Methods for computing weighted pseudoinverses and weighted normal pseudosolutions with singular weights,” Cybern. Syst. Analysis, Vol. 54, No. 3, 398–422 (2018).

    Article  Google Scholar 

  8. S. K. Mitra and C. R. Rao, “Projections under seminorms and generalized Moore–Penroze inverses,” Linear Algebra and its Application, Vol. 9, 155–167 (1974).

    Article  MathSciNet  Google Scholar 

  9. C. R. Rao and S .K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley and Sons, New York (1971).

  10. N. A. Varenyuk, E. F. Galba, I. V. Sergienko, and A. N. Khimich, “Weighted pseudoinversion with indefinite weights,” Ukr. Math. J., Vol 70, No. 6, 866–889 (2018).

    Article  MathSciNet  Google Scholar 

  11. E. F. Galba, V. S. Deineka, and I. V. Sergienko, “Expansions and polynomial limit representations of weighted pseudoinverses,” J. Comp. Math. Math. Physics, Vol. 47, No. 5, 713–731 (2007).

    Article  Google Scholar 

  12. E. F. Galba and N. A. Vareniuk, “Representingn weighted pseudoinverse matrices with mixed weights in terms of other pseudoinverses,” Cybern. Syst. Analysis, Vol. 54, No. 2, 185–192 (2018).

    Article  Google Scholar 

  13. I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products, Birkhauser, Basel–Boston–Stuttgart (1983).

    MATH  Google Scholar 

  14. I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra and Applications, Birkhauser, Basel–Boston–Berlin (2005).

    MATH  Google Scholar 

  15. Kh. D. Ikramov, “Theorem about diagonalization of one type of Hamiltonians from the point of view of the theory of operators in spaces with indefinite metrics,” Zh. Vych. Mat. Mat. Fiz., Vol. 29, No. 1, 3–14 (1989).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Kh. D. Ikramov, “On the algebraic properties of classes of pseudo-permutation and N-self-adjoint matrices,” Zh. Vych. Mat. Mat. Fiz, Vol. 32, No. 8, 1155–1169 (1992).

    MathSciNet  MATH  Google Scholar 

  18. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  19. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press (1992).

    Google Scholar 

  20. I. N. Molchanov and E .F. Galba, “A weighted pseudoinverse for complex matrices,” Ukr. Math. J., Vol. 35, No. 1, 46–50 (1983).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. A. Albert, Regression, Pseudoinversion, and Recurrent Estimation [in Russian], Nauka, Moscow (1977).

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. 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, Vol. 44, No. 6, 863–874 (2008).

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Y. Wei and D. Wang, “Condition numbers and perturbation of the weighted Moore–Penrose inverse and weighted linear least squares problem,” Appl. Math. Comput., Vol. 145, 45–58 (2003).

    MathSciNet  MATH  Google Scholar 

  28. Y. Wei, “A note on the sensitivity of the solution of the weighted linear least squares problem,” Appl. Math. Comput., Vol. 145, 481–485 (2003).

    MathSciNet  MATH  Google Scholar 

  29. A. N. Khimich, A. V. Popov, and V. V. Polyanko, “Algorithms of parallel computations for linear algebra problems with irregularly structured matrices,” Cybern. Syst. Analysis, Vol. 47, No. 6, 973–985 (2011).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    N. A. Vareniuk & N. I. Tukalevska

Authors
  1. N. A. Vareniuk
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  2. N. I. Tukalevska
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Correspondence to N. A. Vareniuk.

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Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2020, pp. 141–151.

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Vareniuk, N.A., Tukalevska, N.I. Existence and Uniqueness of Weighted Pseudoinverses with Nonsingular Indefinite Weights. Cybern Syst Anal 56, 466–475 (2020). https://doi.org/10.1007/s10559-020-00262-5

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  • DOI: https://doi.org/10.1007/s10559-020-00262-5

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