Existence and characterizations of hyper-dual group inverse
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910905498861568 |
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| author | Verma, Tikesh Kumar, Amit Shekhar, Vaibhav |
| author_facet | Verma, Tikesh Kumar, Amit Shekhar, Vaibhav |
| contents | Motivated by the recent work of Xiao and Zhong [AIMS Math. 9 (2024), 35125--35150: MR4840882], we propose a generalized inverse for a hyper-dual matrix called hyper-dual group generalized inverse (HDGGI). Under certain necessary and sufficient conditions, we establish the existence of the HDGGI of a hyper-dual matrix. We then show that the HDGGI is unique (whenever exists). The HDGGI is then used to solve a linear hyper-dual system. We also exploit some sufficient conditions under which the reverse and forward-order laws for a particular form of the HDGGI and HDMPGI hold. We also discuss the least-squares properties of hyper-dual group inverse. Using the definition of dual matrix of order $n$, we finally establish necessary and sufficient condition for the existence of the group inverse of a dual matrix of order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and characterizations of hyper-dual group inverse Verma, Tikesh Kumar, Amit Shekhar, Vaibhav Rings and Algebras Commutative Algebra Functional Analysis 15A09, 15A57, 15A24 Motivated by the recent work of Xiao and Zhong [AIMS Math. 9 (2024), 35125--35150: MR4840882], we propose a generalized inverse for a hyper-dual matrix called hyper-dual group generalized inverse (HDGGI). Under certain necessary and sufficient conditions, we establish the existence of the HDGGI of a hyper-dual matrix. We then show that the HDGGI is unique (whenever exists). The HDGGI is then used to solve a linear hyper-dual system. We also exploit some sufficient conditions under which the reverse and forward-order laws for a particular form of the HDGGI and HDMPGI hold. We also discuss the least-squares properties of hyper-dual group inverse. Using the definition of dual matrix of order $n$, we finally establish necessary and sufficient condition for the existence of the group inverse of a dual matrix of order $n$. |
| title | Existence and characterizations of hyper-dual group inverse |
| topic | Rings and Algebras Commutative Algebra Functional Analysis 15A09, 15A57, 15A24 |
| url | https://arxiv.org/abs/2504.05264 |