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arXiv · 2604.21818

Formulae for the Drazin inverse of Modified Tensors via the Einstein Product

Yue Zhao·Daochang Zhang·Dijana Mosic
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Abstract

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.

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Yue Zhao, Daochang Zhang, Dijana Mosic. 2026-04-23. Formulae for the Drazin inverse of Modified Tensors via the Einstein Product. https://arxiv.org/abs/2604.21818

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