Dijana Mosic

Yue Zhao, Daochang Zhang, Jingqian Li et al.

The motivation of this paper is to investigate the perturbation theory for the QT-Drazin inverse of quaternion tensors under the QT-product via the associated $z$-block circulant representation. A fundamental relationship between the QT-Drazin inverse of $\mathtt{bcirc}_z(\mathcal A)$ and the $z$-block circulant form of $\mathcal A^D$ is established. Moreover, the QT-index of a quaternion tensor is characterized by the indices of the diagonal blocks in the corresponding block-diagonalized matrix. As a consequence, a representation of the QT-Drazin inverse in terms of the QT-Moore--Penrose inverse is derived, which offers a practical approach for its direct computation in MATLAB. Furthermore, a decomposition theory for the QT-Drazin inverse is developed by combining the structure of $z$-block circulant matrices with the Jordan decomposition of quaternion matrices. Numerical examples are provided to demonstrate the theoretical results and computational feasibility.

Yue Zhao, Daochang Zhang, Dijana Mosic

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.