Formulae for the Drazin inverse of Modified Tensors via the Einstein Product
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914502646169600 |
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| author | Zhao, Yue Zhang, Daochang Mosic, Dijana |
| author_facet | Zhao, Yue Zhang, Daochang Mosic, Dijana |
| contents | This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21818 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Formulae for the Drazin inverse of Modified Tensors via the Einstein Product Zhao, Yue Zhang, Daochang Mosic, Dijana Numerical Analysis This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression. |
| title | Formulae for the Drazin inverse of Modified Tensors via the Einstein Product |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.21818 |