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Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 8, Pages 1347–1363 (Mi zvmmf4730)  

This article is cited in 8 scientific papers (total in 8 papers)

Weighted pseudoinverses and weighted normal pseudosolutions with singular weights

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

Institute of Cybernetics, National Academy of Sciences of Ukraine, pr. Akademika Glushkova 40, Kiev, 03680, Ukraine

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.

Key words: weighted pseudoinverses with singular weights, weighted normal pseudosolutions, matrix power series, matrix power products, iterative methods.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:8, 1281–1297

Bibliographic databases:

UDC: 512.61
Received: 22.12.2008

Citation: E. F. Galba, V. S. Deineka, I. V. Sergienko, “Weighted pseudoinverses and weighted normal pseudosolutions with singular weights”, Zh. Vychisl. Mat. Mat. Fiz., 49:8 (2009), 1347–1363; Comput. Math. Math. Phys., 49:8 (2009), 1281–1297

Citation in format AMSBIB
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\paper Weighted pseudoinverses and weighted normal pseudosolutions with singular weights
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\vol 49
\issue 8
\pages 1347--1363
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\transl
\jour Comput. Math. Math. Phys.
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\vol 49
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\pages 1281--1297
\crossref{https://doi.org/10.1134/S0965542509080016}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Sergienko I.V., Galba E.F., Deineka V.S., “Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights”, Ukrainian Math. J., 63:1 (2011), 98–124  crossref  mathscinet  zmath  isi  elib  scopus
    2. Sergienko I.V., Deineka V.S., “Numerical solution to some inverse problems for elliptic systems using pseudoinverse matrices”, Journal of Automation and Information Sciences, 43:7 (2011), 3–28  crossref  mathscinet  isi  scopus
    3. E. F. Galba, V. S. Deineka, I. V. Sergienko, “Vzveshennoe singulyarnoe razlozhenie i vzveshennoe psevdoobraschenie matrits s vyrozhdennymi vesami”, Zh. vychisl. matem. i matem. fiz., 52:12 (2012), 2115–2132  mathnet
    4. Galba E.F. Deineka V.S. Sergienko I.V., “Necessary and Sufficient Conditions for the Existence of a Weighted Singular Value Decomposition of Matrices with Singular Weights”, Dokl. Math., 89:2 (2014), 182–184  crossref  mathscinet  zmath  isi  elib  scopus
    5. Sergienko I.V. Galba E.F. Deineka V.S., “Necessary and Sufficient Conditions For the Existence of Weighted Singular-Valued Decompositions of Matrices With Singular Weights”, Ukr. Math. J., 67:3 (2015), 464–486  crossref  mathscinet  zmath  isi  elib  scopus
    6. Sergienko I.V. Galba E.F., “Weighted Singular Value Decomposition of Matrices With Singular Weights Based on Weighted Orthogonal Transformations”, Cybern. Syst. Anal., 51:4 (2015), 514–528  crossref  mathscinet  zmath  isi  elib  scopus
    7. Sergienko I.V. Galba E.F., “Weighted Pseudoinversion with Singular Weights”, Cybern. Syst. Anal., 52:5 (2016), 708–729  crossref  mathscinet  zmath  isi
    8. Galba E.F., Sergienko I.V., “Methods For Computing Weighted Pseudoinverses and Weighted Normal Pseudosolutions With Singular Weights”, Cybern. Syst. Anal., 54:3 (2018), 398–422  crossref  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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