arXiv subjects

Qing-Wen Wang

Publications and source records attributed to Qing-Wen Wang.

At least 19 recordsLinked to original sources

arXiv2025

Further properties and representations of the W-weighted m-weak group inverse

The purpose of this paper is to explore more properties and representations of the W-weighted m-weak group (in short, W-m-WG) inverse. We first explore an interesting relation between two projectors with respect to the W-m-WG inverse. Then, the W-m-WG inverse is represented by various generalized inverses including W-weighted Drazin inverse, W-weighted weak group inverse, W-weighted core inverse, etc. We also give three concise explicit expressions for the W-m-WG inverse. Moreover, a canonical form of the W-m-WG inverse is presented in terms of the singular value decomposition. Finally, several numerical examples are designed to illustrate some results given in the paper.

math.RA↗

arXiv2024

The W-weighted m-weak group MP inverse and its applications

We extend the concept of the m-weak group MP inverse of a square matrix to a rectangular matrix, called the W-weighted m-weak group MP inverse, which also unifies the W-weighted weak core inverse and W-weighted DMP inverse. Some properties, characterizations and representations of this new generalized inverse are shown. Additionally, applications of the W-weighted weak group MP inverse are given in solving a constrained optimization problem and a class of consistent matrix equations.

math.FA↗

arXiv2024

On Rayleigh Quotient Iteration for Dual Quaternion Hermitian Eigenvalue Problem

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing minimal residual property of the Rayleigh Quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.

math.NA↗

arXiv2024

QQMR: A Structure-Preserving Quaternion Quasi-Minimal Residual Method for Non-Hermitian Quaternion Linear Systems

The quaternion biconjugate gradient (QBiCG) method, as a novel variant of quaternion Lanczos-type methods for solving the non-Hermitian quaternion linear systems, does not yield a minimization property. This means that the method possesses a rather irregular convergence behavior, which leads to numerical instability. In this paper, we propose a new structure-preserving quaternion quasi-minimal residual method, based on the quaternion biconjugate orthonormalization procedure with coupled two-term recurrences, which overcomes the drawback of QBiCG. The computational cost and storage required by the proposed method are much less than the traditional QMR iterations for the real representation of quaternion linear systems. Some convergence properties of which are also established. Finally, we report the numerical results to show the robustness and effectiveness of the proposed method compared with QBiCG.

math.NA↗

arXiv2023

A system of dual quaternion matrix equations with its applications

We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for solving a system of the dual quaternion matrix equations $(AX, XC) = (B, D)$, along with providing an expression for its general solution. Serving as an application, we investigate the solutions to the dual quaternion matrix equations $AX = B$ and $XC=D$, including $η$-Hermitian solutions. Lastly, we design a numerical example to validate the main research findings of this paper.

math.RA↗

arXiv2023

Gl-QFOM and Gl-QGMRES: two efficient algorithms for quaternion linear systems with multiple right-hand sides

In this paper, we propose the global quaternion full orthogonalization (Gl-QFOM) and global quaternion generalized minimum residual (Gl-QGMRES) methods, which are built upon global orthogonal and oblique projections onto a quaternion matrix Krylov subspace, for solving quaternion linear systems with multiple right-hand sides. We first develop the global quaternion Arnoldi procedure to preserve the quaternion Hessenberg form during the iterations. We then establish the convergence analysis of the proposed methods, and show how to apply them to solve the Sylvester quaternion matrix equation. Numerical examples are provided to illustrate the effectiveness of our methods compared with the traditional Gl-FOM and Gl-GMRES iterations for the real representations of the original linear systems.

math.NA↗

arXiv2023

Algebraic conditions and general solution to a system of quaternion tensor equations with applications

This paper investigates the necessary and sufficient algebraic conditions to a constrained system of Sylvester-type quaternion tensor equations. An explicit formula of the general solution regarding the Moore-Penrose inverses of some block given tensors is obtained. As an application of a particular case, we establish the solvability conditions and the general solution to a system of Sylvester-type quaternion tensor equations involving $η$-Hermitian unknowns. An algorithm with a numerical example is proposed to compute the general solution of the main system.

math.RA↗

arXiv2023

A new generalization of a system of two-sided coupled Sylvester-like quaternion tensor equations

This study establishes consistency conditions and a general solution for a coupled system that consists of five two-sided Sylvester-like tensor equations in ten quaternion variables throughout the Einstein tensor product. Certain specific cases are thus established. In a direct application, we investigate certain necessary and sufficient conditions for the existence of an $η$-Hermitian solution to five coupled two-sided Sylvester-like quaternion tensor equations. Finally, we present an algorithm and a numerical example to validate the main result.

math.RA↗

arXiv2022

The general solutions to some systems of Sylvester-type quaternion matrix equations with an application

Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion matrix equations to be consistent and derive an expression of its general solution (when it is solvable). As an application, we investigate the necessary and sufficient conditions for quaternion matrix equations to be consistent and derive a formula for its general solution involving $η$-Hermicity. As a special case, we also present the necessary and sufficient conditions for the system of two-sided Sylvester-type quaternion matrix equations to have a solution and derive a formula for its general solution (when it is solvable). Finally, we present an algorithm and an example to illustrate the main results of this paper.

math.RA↗

arXiv2022

A Sylvester-type matrix equation over the Hamilton quaternions with an application

We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion matrix equation, which involves $η$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.

math.RA↗

arXiv2020

A system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables

In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables $A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}Z_{i+1}G_{i}=E_{i},~i=\overline{1,k}$. As applications of this system, we present rank equalities as the necessary and sufficient conditions for the existence of a general solution to some systems of quaternion matrix equations $A_{i}X_{i}+(A_{i}X_{i})_ϕ+C_{i}Z_{i}(C_{i})_ϕ+F_{i}Z_{i+1}(F_{i})_ϕ=E_{i},~i=\overline{1,k}$.

math.RA↗

arXiv2020

The solvability conditions and exact solutions to some quaternion tensor systems

We derive necessary and sufficient conditions for the existence of the exact solution to the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{X}_i+ \mathcal{Y}_i\ast_{M}\mathcal{B}_i+\mathcal{C}_i\ast_{N} \mathcal{Z}_i\ast_{M}\mathcal{D}_i+\mathcal{F}_i\ast_{N} \mathcal{Z}_{i+1}\ast_{M}\mathcal{G}_i=\mathcal{E}_i, i=\overline{1,3} $ using Moore-Penrose inverse, and present an expression of the general solution to the system when it is solvable. As an application of this system, we provide the solvability conditions and general solutions for the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{Z}_i\ast_{M}\mathcal{B}_i+ \mathcal{C}_i\ast_{N}\mathcal{Z}_{i+1}\ast_{M}\mathcal{D}_i= \mathcal{E}_i, i=\overline{1,4}. $ This paper can also serve as extensions to some known results.

math-ph↗

arXiv2020

Characterizations of annihilator $(b,c)$-inverses in arbitrary rings

In this paper, we investigate some properties of annihilator $(b,c)$-inverses in an arbitrary ring. We demonstrate that one-sided annihilator $(b,c)$-inverses of elements in arbitrary rings may behave differently in contrast to one-sided $(b,c)$-inverses. Also, we discuss intertwining property, absorption law, reverse order law, and Cline's formula for annihilator $(b,c)$-inverses. As applications, we improve and extend some known results to $(b,c)$-inverses. In particular, we derive an equivalent condition of intertwining property for $(b,c)$-inverses in semigroups.

math.RA↗

arXiv2018

The permanent functions of tensors

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors.

math.CO↗

arXiv2017

Tensor decompositions and tensor equations over quaternion algebra

In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, quaternion tensor rank decomposition, and $η$-Hermitian quaternion tensor decomposition with the isomorphic group structures and Einstein product. Then we give the expression of the Moore-Penrose inverse of a quaternion tensor by using the quaternion tensor SVD. Moreover, we consider a generalized Sylvester quaternion tensor equation. We give some necessary and sufficient conditions for the existence of a solution to the generalized Sylvester quaternion tensor equation in terms of the Moore-Penrose inverses of the quaternion tensors. We also present the expression of the general solution to this tensor equation when it is solvable. As applications of this generalized Sylvester quaternion tensor equation, we derive some necessary and sufficient conditions for the existences of $η$-Hermitian solutions to some quaternion tensor equations. We also provide some numerical examples to illustrate our results.

math.RA↗

arXiv2017

Systems of four coupled one sided Sylvester-type real quaternion matrix equations and their applications

In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.

math.RA↗

arXiv2017

A simultaneous decomposition of four real quaternion matrices encompassing $η$-Hermicity and its applications

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. Let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For $η\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix $A$ is said to be $η$-Hermitian if $A^{η*}=A$ where $A^{η*}=-ηA^{\ast}η$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A=A^{η*}\in \mathbb{H}^{m\times m}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}}$. As applications of this simultaneous matrix decomposition, we derive necessary and sufficient conditions for some real quaternion matrix equations involving $η$-Hermicity in terms of ranks of the coefficient matrices. We also present the general solutions to these real quaternion matrix equations. Moreover, we provide some numerical examples to illustrate our results.

math.RA↗

arXiv2017

The complete equivalence canonical form of four matrices over an arbitrary division ring

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.

math.RA↗