The Perturbation Bound for the T-Drazin Inverse of Tensor and its Application

Abstract

In this paper, let $\mathcal{A}$ and $\mathcal{B}$ be $n\times n\times p$ complex tensors and $\mathcal{B}=\mathcal{A}+\mathcal{E}$. Denote the T-Drazin inverse of $\mathcal{A}$ by $\mathcal{A}^{D}$. We give a perturbation bound for $\|\mathcal{B}^{D}-\mathcal{A}^{D}\|/\|\mathcal{A}^{D}\|$ under condition ($\mathcal{W}$). Considering the solution of singular tensor equation $\mathcal{A}\ast x=b,\ (b\in \mathcal{R}(\mathcal{A}^{D}))$ at the same time. Therefore, the optimal perturbation of T-Drazin inverse of tensors and the solution of a system of tensor equations have been given.