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

Yue Zhao, Daochang Zhang, Jingqian Li, Dijana Mosic

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.

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

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 -block circulant representation. A fundamental relationship between the QT-Drazin inverse of and the -block circulant form of 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 -block circulant matrices with the Jordan decomposition of quaternion matrices. Numerical examples are provided to demonstrate the theoretical results and computational feasibility.
Paper Structure (5 sections, 19 theorems, 112 equations)

This paper contains 5 sections, 19 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and key lemmas
  3. QT-Drazin inverse of $\mathtt{bcirc_z}(\mathcal{A})$
  4. Perturbation analysis of the QT-Drazin inverse
  5. Numerical example

Key Result

Lemma 2.10

DY2026 Let $\mathcal{A}_{\mathbf{d}}$ and $\mathcal{A}_{\mathbf{c}}\in \mathbb C^{n_1\times r\times n_3}$ satisfy that $\mathcal{A}=\mathcal{A}_{\mathbf{d}} + \mathbf{j} \mathcal{A}_{\mathbf{c}}\in\mathbb{Q}^{{n_1\times r\times n_3}}.$ Denote that $\hat{\mathcal{A}}$ is the DFT of $\mathcal{A}$. The

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 34 more