Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
Given a square matrix AA, we are able to construct numerous equalities involving reasonable mixed operations of AA and its conjugate transpose A∗{A}^{\ast }, Moore-Penrose inverse A†{A}^{\dagger } and group inverse A#{A}^{\#}. Such kind of equalities can be generally represented in the equation form...
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De Gruyter
2025-05-01
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| Series: | Demonstratio Mathematica |
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| Online Access: | https://doi.org/10.1515/dema-2024-0089 |
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| author | Tian Yongge |
| author_facet | Tian Yongge |
| author_sort | Tian Yongge |
| collection | DOAJ |
| description | Given a square matrix AA, we are able to construct numerous equalities involving reasonable mixed operations of AA and its conjugate transpose A∗{A}^{\ast }, Moore-Penrose inverse A†{A}^{\dagger } and group inverse A#{A}^{\#}. Such kind of equalities can be generally represented in the equation form f(A,A∗,A†,A#)=0f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0. In this article, the author constructs a series of simple or complicated matrix equalities composed of AA, A∗{A}^{\ast }, A†{A}^{\dagger }, A#{A}^{\#} and their algebraic operations, as well as established various explicit formulas for calculating the ranks of these matrix expressions. Many applications of these matrix rank equalities are presented, including a broad range of necessary and sufficient conditions for a square matrix to be range-Hermitian and Hermitian/skew-Hermitian, respectively. |
| format | Article |
| id | doaj-art-d09283bcaf4446f7b88e6eddafc9ddd8 |
| institution | OA Journals |
| issn | 2391-4661 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Demonstratio Mathematica |
| spelling | doaj-art-d09283bcaf4446f7b88e6eddafc9ddd82025-08-20T02:01:39ZengDe GruyterDemonstratio Mathematica2391-46612025-05-0158143944310.1515/dema-2024-0089Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodologyTian Yongge0College of Technology and Data, Yantai Nanshan University, Yantai, Shandong, ChinaGiven a square matrix AA, we are able to construct numerous equalities involving reasonable mixed operations of AA and its conjugate transpose A∗{A}^{\ast }, Moore-Penrose inverse A†{A}^{\dagger } and group inverse A#{A}^{\#}. Such kind of equalities can be generally represented in the equation form f(A,A∗,A†,A#)=0f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0. In this article, the author constructs a series of simple or complicated matrix equalities composed of AA, A∗{A}^{\ast }, A†{A}^{\dagger }, A#{A}^{\#} and their algebraic operations, as well as established various explicit formulas for calculating the ranks of these matrix expressions. Many applications of these matrix rank equalities are presented, including a broad range of necessary and sufficient conditions for a square matrix to be range-Hermitian and Hermitian/skew-Hermitian, respectively.https://doi.org/10.1515/dema-2024-0089block matrixgroup inversemoore-penrose inverserangerankreverse-order law15a0915a24 |
| spellingShingle | Tian Yongge Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology Demonstratio Mathematica block matrix group inverse moore-penrose inverse range rank reverse-order law 15a09 15a24 |
| title | Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology |
| title_full | Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology |
| title_fullStr | Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology |
| title_full_unstemmed | Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology |
| title_short | Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology |
| title_sort | towards finding equalities involving mixed products of the moore penrose and group inverses by matrix rank methodology |
| topic | block matrix group inverse moore-penrose inverse range rank reverse-order law 15a09 15a24 |
| url | https://doi.org/10.1515/dema-2024-0089 |
| work_keys_str_mv | AT tianyongge towardsfindingequalitiesinvolvingmixedproductsofthemoorepenroseandgroupinversesbymatrixrankmethodology |