Perturbation Analysis of the QT-Drazin Inverse of Quaternion Tensors via the QT-Product

TL;DR

This paper develops a perturbation theory for the QT-Drazin inverse of quaternion tensors using the QT-product, establishing relationships with block circulant matrices, and provides practical computational methods with numerical demonstrations.

Contribution

It introduces a novel perturbation framework for the QT-Drazin inverse of quaternion tensors, linking it to block circulant representations and enabling efficient MATLAB computations.

Findings

Established a relationship between QT-Drazin inverse and block circulant matrices.

Derived a practical MATLAB computation method for the QT-Drazin inverse.

Provided numerical examples validating the theoretical results.

Abstract

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.