通过QT-积研究四元数张量的QT-德拉兹因逆的扰动分析
Perturbation Analysis of the QT-Drazin Inverse of Quaternion Tensors via the QT-Product
AI总结：
本文通过关联的z-块循环表示研究四元数张量的QT-德拉兹因逆的扰动理论，建立了QT-德拉兹因逆与张量的z-块循环形式之间的基本关系，并提出通过QT-摩尔-彭罗斯逆表示QT-德拉兹因逆的方法，提供MATLAB直接计算的实用途径。
摘要：
本文的动机是通过关联的$z$-块循环表示研究四元数张量在QT-积下的QT-德拉兹因逆的扰动理论。建立了QT-德拉兹因逆与$z$-块循环形式之间的基本关系。此外，四元数张量的QT指数通过对应块对角化矩阵的对角块的指数来表征。作为结果，通过QT-摩尔-彭罗斯逆表示QT-德拉兹因逆的表示被推导出来，这为在MATLAB中直接计算提供了实用方法。进一步地，通过结合$z$-块循环矩阵的结构与四元数矩阵的约当分解，发展了QT-德拉兹因逆的分解理论。数值示例用于展示理论结果和计算可行性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
