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

Xianchun Meng; Ricai Luo; Xingshou Huang; Guiying Wang

doi:doi:10.11648/j.pamj.20221106.13

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

Xianchun Meng, Ricai Luo, Xingshou Huang, Guiying Wang

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 6)

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 A_ij^⊕ 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(A^k), where A∈Cm,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), 2024. Published by Science Publishing Group

Keywords

Core-EP Inverse, Characterizations, Representations, Frobenius Norm

References

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[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.
[13]	Jirgen Groβ. Solution to a rank equation [J]. Linear Algebra and its Appli-cations, 1999, 289 (289): 127-130.

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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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TY  - JOUR
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
PY  - 2022
N1  - https://doi.org/10.11648/j.pamj.20221106.13
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
SP  - 112
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

Xianchun Meng

Department of Mathematics and Physics, Hechi University, YiZhou, China
Ricai Luo

Department of Mathematics and Physics, Hechi University, YiZhou, China
Xingshou Huang

Department of Mathematics and Physics, Hechi University, YiZhou, China
Guiying Wang

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

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
PY  - 2022
N1  - https://doi.org/10.11648/j.pamj.20221106.13
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
SP  - 112
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  -

Copy | Download