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Daochang Zhang

Publications and source records attributed to Daochang Zhang.

11 recordsLinked to original sources
arXiv2026

Formulae for the Drazin inverse of Modified Tensors via the Einstein Product

This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression.

math.NA↗
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

On the Dual Drazin Inverse of Adjacency Matrices of Dual-number-Weighted Digraphs

The motivation of this paper is to investigate the dual Drazin inverse of adjacency matrices arising from several classes of connected dual-number-weighted digraphs over the dual complex algebra. Explicit formulas for the dual Drazin inverse of dual complex anti-triangular block matrices are derived under suitable assumptions. These results are applied to DN-DS digraphs, DN-DLS digraphs, and DN-DW digraphs. In particular, the assumptions in \cite{AMPMJM2026} are weakened for DN-DS digraphs, an open problem in \cite{AMPMJM2026} for the case $BC=0$ is generalized and solved for DN-DLS digraphs. And the group inverse result in \cite{MNSEJAA2022} for bipartite block form adjacency matrices is extended to the dual Drazin inverse for DN-DW digraphs. We further derive explicit formulas for the dual group inverse and dual Drazin inverse of another adjacency matrix for DN-DW digraphs.

math.CO↗
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↗
arXiv2019

Weighted weak group inverse for Hilbert space operators

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, introduced to extend weak group inverse for square matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.

math.FA↗
arXiv2018

On the existence of group inverses of Peirce corner matrices

We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a $2 \times 2$ block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the $2 \times 2$ block matrix are obtained and some of them generalize several results in the literature.

math.RA↗
arXiv2017

Drazin inverses of the sum of four matrices and its applications

Our aim is to establish relations between Drazin inverses of the pesudo-block matrix $(P,Q,R,S)$ and the block matrix composed of $P,R,S,Q$, where $R^2=S^2=0$. Based on the relations, we give representations for Drazin inverses of the sum $P+Q+R+S$ under weaker restrictions. As its applications, several expressions for Drazin inverses of a $2\times 2$ block matrix are presented under some assumptions. Our results generalize several results in the literature.

math.RA↗
arXiv2013

Representations for the Drazin inverse of the generalized Schur complement

In this paper we present expressions for the Drazin inverse of the generalized Schur complement $A-CD^{d}B$ in terms of the Drazin inverses of $A$ and the generalized Schur complement $D-BA^{d}C$ under less and weaker restrictions, which generalize several results in the literature and the formula of Sherman-Morrison-Woodbury type.

math.RA↗