arXiv subjects

Jingqian Li

Publications and source records attributed to Jingqian Li.

3 recordsLinked to original sources

arXiv2026

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

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.

math.NA↗

arXiv2026

Perturbation analysis of tensor $(\mathcal{B},\mathcal{C})$-inverse via Einstein product

We investigate the influence of a relatively small perturbation on various generalized inverses functions or quantities derived from a tensor $\mathcal{A}$.When a small tensor perturbation $\mathcal{E}$ is introduced, it becomes challenging to analyze generalized inverses of the perturbed tensor $ \mathcal{D} =\mathcal{A}+\mathcal{E}$ and to determine how this perturbation affects a generalized inverse of $\mathcal{A}$.Our main goal is to understand the relationship between $\mathcal{D}^\Game$ and $ \mathcal{A}^\Game $, where $(\cdot)^\Game$ denotes a specific generalized inverse or a class of generalized inverses.In particular, classes of tensor inner, outer, and $(\mathcal{B},\mathcal{C})$ inverses are considered.

math.FA↗

arXiv2026

On the Block-Diagonalization and Multiplicative Equivalence of Quaternion $Z$-Block Circulant Matrices with their Applications

The motivation of this paper is twofold. First, we investigate the block-diagonalization of the $z$-block circulant matrix $\mathtt{bcirc_z}(\mathcal A)$, based on this block-diagonal structure, and develop the algorithm $\mathtt{bcirc_z}$-inv for computing the inverse of $\mathtt{bcirc_z}(\mathcal A)$. Second, we establish the equivalence between the QT-product of tensors and the product of the corresponding $z$-block circulant matrices. Based on this equivalence and in combination with the algorithm $\mathtt{bcirc_z}$-inv, large-scale tests and scalability analysis of the Tikhonov-regularized model are conducted. As a by-product of the analysis, some relevant and straightforward properties of the quaternion $z$-block circulant matrices are provided. As applications, a series of quaternion tensor decompositions under the QT-product and their corresponding $z$-block circulant matrices decompositions are obtained, including the QT-Polar decomposition, the QT-PLU decomposition, and the QT-LU decomposition. Meanwhile, the QT-SVD is rederived based on the relation between $\mathcal A$ and $\mathtt{bcirc_z}(\mathcal A)$. Furthermore, we develop corresponding algorithms and present several large-scale tests and scalability analysis. In addition, applications in video rotation are presented to evaluate several rotation strategies based on the QT-Polar decomposition, which shows the decomposition remains stable and inter-frame consistent while accurately maintaining color reproduction.

math.NA↗