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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-05-01
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| Series: | Demonstratio Mathematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/dema-2024-0089 |
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| Summary: | 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. |
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| ISSN: | 2391-4661 |