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

Abstract

This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal{A}-\mathcal{C}{\ast }_N{\mathcal{D}}^D{\ast }_N\mathcal{B}$ via the Einstein product, formulated using the Drazin inverse of $\mathcal{A}$ and the generalized Schur complement $\mathcal{D}-\mathcal{B}{\ast }_N{\mathcal{A}}^D{\ast }_N\mathcal{C}$, 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.