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Characterizations and Representations of the Core-EP Inverse and Its Applications

Received: 13 October 2022     Accepted: 4 November 2022     Published: 30 November 2022
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Abstract

Generalized inverse matrices are an important branch of matrix theory, have a wide range of applications in many fields, such as mathematical statistics, system theory, optimization computing and cybernetics etc. This paper mainly studies the correlation properties and applications of the Core-ep inverse. Firstly, we present the characterizations of the Core-EP inverse by the matrix equations, and an example is given for analysis. Secondly, we present a representation for computing the Core-EP inverse, get a representation of Aij by Cramer rule , and an example is given for analysis. Finally, we study the constrained matrix approximation problem in the Frobenius norm by using the Core-EP inverse: ║Ax-bF=min subject to xR(Ak), where AC m,m , we obtain the unique solution to the problem.

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 6)
DOI 10.11648/j.pamj.20221106.13
Page(s) 112-120
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Core-EP Inverse, Characterizations, Representations, Frobenius Norm

References
[1] Wang G, Wei Y, Qiao S. Generalized inverses: theory and computations [M]. Singapore, Beijing: Springer, Science Press; 2018. (Developments in Mathematics; 53).
[2] Campbell S L, Meyer C D. Generalized Inverses of Linear Transforma-tions [M] Generalized inverses of linear transformations. Pitman, 1979.
[3] Prasad K M, Mohana K S. Core - EP inverse [J]. Linear Multilinear Alge-bra, 2014, 62 (6): 792-802.
[4] Drazin, M. P. Pseudo-Inverses in Associative Rings and Semigroups [J]. The American Mathematical Monthly, 1958, 65 (7): 506.
[5] Keiichi M, Rozloznik Miroslav. On GMRES for Singular EP and GPSystems [J]. SIAM Journal on Matrix Analysis and Applications, 2018, 39 (2): 1033-1048.
[6] H Wang, Zhang X. The core inverse and constrained matrix approximation problem [J]. Open Mathematics, 2020, 18 (1): 653-661.
[7] Fiedler M, Markham T L. A characterization of the Moore- Penrose in-verse [J]. Linear Algebra and Its Applications, 1993, 179 (1): 129-133.
[8] Wei Y A characterization and representation of the Drazin inverse [J]. SIAMJ matrix Anal Appl, 1996; 17: 744-747.
[9] Ma H, Li T. Characterizations and representations of the core inverse and its applications [J]. Linear and Multilinear Algebra, 2019: 1-11.
[10] Ma H, Stanimirovi P S. Characterizations, approximation and pertur-bations of the core-EP inverse [J]. Applied Mathematics and Computation, 2019, 359.
[11] Wang H. Core-EP decomposition and its applications [J]. Linear Algebra and Its Applications, 2016, 508: 289- -300.
[12] Yuan Y, Zuo K. Compute limx- →oX (XIp + YAX)-1Y by the prod-uct singular value decomposition [J]. Linear and Multilinear Algebra, 2016; 64: 269-278.
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  • APA Style

    Xianchun Meng, Ricai Luo, Xingshou Huang, Guiying Wang. (2022). Characterizations and Representations of the Core-EP Inverse and Its Applications. Pure and Applied Mathematics Journal, 11(6), 112-120. https://doi.org/10.11648/j.pamj.20221106.13

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    ACS Style

    Xianchun Meng; Ricai Luo; Xingshou Huang; Guiying Wang. Characterizations and Representations of the Core-EP Inverse and Its Applications. Pure Appl. Math. J. 2022, 11(6), 112-120. doi: 10.11648/j.pamj.20221106.13

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    AMA Style

    Xianchun Meng, Ricai Luo, Xingshou Huang, Guiying Wang. Characterizations and Representations of the Core-EP Inverse and Its Applications. Pure Appl Math J. 2022;11(6):112-120. doi: 10.11648/j.pamj.20221106.13

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  • @article{10.11648/j.pamj.20221106.13,
      author = {Xianchun Meng and Ricai Luo and Xingshou Huang and Guiying Wang},
      title = {Characterizations and Representations of the Core-EP Inverse and Its Applications},
      journal = {Pure and Applied Mathematics Journal},
      volume = {11},
      number = {6},
      pages = {112-120},
      doi = {10.11648/j.pamj.20221106.13},
      url = {https://doi.org/10.11648/j.pamj.20221106.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221106.13},
      abstract = {Generalized inverse matrices are an important branch of matrix theory, have a wide range of applications in many fields, such as mathematical statistics, system theory, optimization computing and cybernetics etc. This paper mainly studies the correlation properties and applications of the Core-ep inverse. Firstly, we present the characterizations of the Core-EP inverse by the matrix equations, and an example is given for analysis. Secondly, we present a representation for computing the Core-EP inverse, get a representation of Aij⊕ by Cramer rule , and an example is given for analysis. Finally, we study the constrained matrix approximation problem in the Frobenius norm by using the Core-EP inverse: ║Ax-b║F=min subject to x∈R(Ak), where A∈C m,m , we obtain the unique solution to the problem.},
     year = {2022}
    }
    

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    T1  - Characterizations and Representations of the Core-EP Inverse and Its Applications
    AU  - Xianchun Meng
    AU  - Ricai Luo
    AU  - Xingshou Huang
    AU  - Guiying Wang
    Y1  - 2022/11/30
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    DO  - 10.11648/j.pamj.20221106.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    EP  - 120
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20221106.13
    AB  - Generalized inverse matrices are an important branch of matrix theory, have a wide range of applications in many fields, such as mathematical statistics, system theory, optimization computing and cybernetics etc. This paper mainly studies the correlation properties and applications of the Core-ep inverse. Firstly, we present the characterizations of the Core-EP inverse by the matrix equations, and an example is given for analysis. Secondly, we present a representation for computing the Core-EP inverse, get a representation of Aij⊕ by Cramer rule , and an example is given for analysis. Finally, we study the constrained matrix approximation problem in the Frobenius norm by using the Core-EP inverse: ║Ax-b║F=min subject to x∈R(Ak), where A∈C m,m , we obtain the unique solution to the problem.
    VL  - 11
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematics and Physics, Hechi University, YiZhou, China

  • Department of Mathematics and Physics, Hechi University, YiZhou, China

  • Department of Mathematics and Physics, Hechi University, YiZhou, China

  • Department of Mathematics and Physics, Hechi University, YiZhou, China

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