^{1}Khimich, AN, ^{1}Galba, EF, ^{1}Vareniuk, NA^{1}V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv |

Dopov. Nac. akad. nauk Ukr. 2017, 6:14-20 |

https://doi.org/10.15407/dopovidi2017.06.014 |

Section: Mathematics |

Language: Russian |

Abstract: Weighted pseudoinverse matrices with nonsingular indefinite weights are defined and analyzed. The theorem of existence and uniqueness of these matrices is proved. A representation of weighted pseudoinverse matrices with indefinite weights is given in terms of coefficients of the characteristic polynomials of symmetrizable matrices. Their expansions in matrix power series or products are obtained. The limiting representations of those matrices are obtained. |

Keywords: limiting representations of weighted pseudoinverse matrices, matrix power series and products, weighted pseudoinverse matrices with nonsingular indefinite weights |

References:

- Chipman, J. S. (1964). On least squares with insufficient observation. J. Amer.Statist. Assoc., 59, No. 308, pp. 1078-1111.
- Milne, R. D. (1968). An oblique matrix pseudoinverse. SIAM J. Appl. Math., 16, No. 5, pp. 931-944.
- Ward, J. F., Boullion, T. L. & Lewis, T. O. (1971). A note on the oblique matrix pseudoinverse. SIAM J. Appl. Math., 20, No. 2, pp. 173-175.
- Ward, J. F., Boullion, T. L. & Lewis, T.O. (1971). Weighted pseudoinverses with singular weights. SIAM J. Appl. Math., 21, No. 3, pp. 480-482.
- Galba, E. F., Deineka, V. S. & Sergienko, I. V. (2009). Weighted pseudoinverses and weighted normal pseudosolutions with singular weights. Comput. Math. Math. Phys., 49, No. 8, rr. 1281-1297.
- Sergienko, I. V., Galba, E. F. & Deineka, V. S. (2011). Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights. Ukr. Math. J., 63, Art. 98.
- Sergienko, I. V., Galba, Y. F. & Deineka, V. S. (2011). Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights. Cybern. Syst. Anal., 47, Iss. 1, pp. 11-28.
- Mitra, S. K. & Rao, C. R. (1974). Projections under seminorms and generalized Moore—Penroze inverses. Linear Algebra Appl., No. 9, pp. 155-167.
- Censor, Y. & Elfving, T. (2002). Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem. SIAM J. Matrix. Anal. Appl., 24, No. 1, pp. 40-58.
- Censor, Y. & Elfving, T. (2003). Iterative algorithms with seminorm-induced oblique projections. Abstr. Appl. Anal., No. 7, pp. 387-406.
- Khimich, A.N. & Nikolaevskaya, E.A. (2008). Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data. Cybern. Syst. Anal., 44, Iss. 6, pp. 863-874.
- Nikolaevskaya, E.A. & Khimich, A.N. (2009). Error estimation for a weighted minimum-norm least squares solution with positive definite weights. Comput. Math. Math. Phys., 49, Iss. 3, pp. 409-417.