Formulae for the Drazin inverse of Modified Tensors via the Einstein Product
Provides theoretical generalizations of matrix inversion formulas to tensor algebra, but the contribution is incremental as it extends known results to higher-order tensors without demonstrating practical impact.
This paper derives exact expressions for the Drazin inverse of a modified tensor under the Einstein product, generalizing and unifying existing results for second-order tensors and reducing to the classical Sherman-Morrison-Woodbury formula. An illustrative example is provided.
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.