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Symmetry
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Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations
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Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 May 2025 | Viewed by 5631

Special Issue Editor

Prof. Dr. Qing-Wen Wang

E-Mail Website
Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: Matrix Theory; linear and multilinear algebra; numerical linear algebra
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry plays a crucial role in various mathematical fields, including in the study of matrices and equations. Dual quaternions, an extension of the concept of quaternions, are a powerful mathematical tool used in computer graphics, robotics, and physics.

Symmetry plays a crucial role in the study of dual quaternion matrices and matrix equations. Dual quaternions, as mathematical objects, possess inherent symmetries that reflect their geometric interpretations and allow for efficient computations. These symmetries can be exploited to simplify matrix equations and to develop more efficient algorithms for solving them.

By studying the symmetries of dual quaternion matrices within matrix equations, researchers can uncover deeper connections between geometric transformations and algebraic structures. This interdisciplinary approach paves the way for innovative applications in fields such as computer animation, robotic control, and mathematical physics.

Prof. Dr. Qing-Wen Wang
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • dual quaternion matrices
  • matrix equations
  • matrix symmetry
  • symmetric matrices
  • orthogonal matrices
  • unitary matrices
  • matrix equations
  • Hermitian matrices
  • skew-symmetric matrices
  • Cayley transform
  • Lie groups and Lie algebras

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Published Papers (6 papers)

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Research

22 pages, 350 KiB  
Article
The Right–Left WG Inverse Solutions to Quaternion Matrix Equations
by Ivan Kyrchei, Dijana Mosić and Predrag Stanimirović
Symmetry 2025, 17(1), 38; https://doi.org/10.3390/sym17010038 - 28 Dec 2024
Viewed by 499
Abstract
This paper studies new characterizations and expressions of the weak group (WG) inverse and its dual over the quaternion skew field. We introduce a dual to the weak group inverse for the first time in the literature and give some new characterizations for [...] Read more.
This paper studies new characterizations and expressions of the weak group (WG) inverse and its dual over the quaternion skew field. We introduce a dual to the weak group inverse for the first time in the literature and give some new characterizations for both the WG inverse and its dual, named the right and left weak group inverses for quaternion matrices. In particular, determinantal representations of the right and left WG inverses are given as direct methods for their constructions. Our other results are related to solving the two-sided constrained quaternion matrix equation AXB=C and the according approximation problem that could be expressed in terms of the right and left WG inverse solutions. Within the framework of the theory of noncommutative row–column determinants, we derive Cramer’s rules for computing these solutions based on determinantal representations of the right and left WG inverses. A numerical example is given to illustrate the gained results. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
18 pages, 339 KiB  
Article
Symmetric Functions and Rings of Multinumbers Associated with Finite Groups
by Yurii Chopiuk and Andriy Zagorodnyuk
Symmetry 2025, 17(1), 33; https://doi.org/10.3390/sym17010033 - 27 Dec 2024
Viewed by 636
Abstract
In this paper, we introduce ωn-symmetric polynomials associated with the finite group ωn, which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces ℓ1. These polynomials extend the classical [...] Read more.
In this paper, we introduce ωn-symmetric polynomials associated with the finite group ωn, which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces ℓ1. These polynomials extend the classical notions of symmetric and supersymmetric polynomials on ℓ1. We explore algebraic bases in the algebra of ωn-symmetric polynomials and derive corresponding generating functions. Building on this foundation, we construct rings of multisets (multinumbers), defined as equivalence classes on the underlying space under the action of ωn-symmetric polynomials, and investigate their fundamental properties. Furthermore, we examine the ring of integer multinumbers associated with the group ωn, proving that it forms an integral domain when n is prime or n=4. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
13 pages, 271 KiB  
Article
Solving the Dual Generalized Commutative Quaternion Matrix Equation AXB = C
by Lei Shi, Qing-Wen Wang, Lv-Ming Xie and Xiao-Feng Zhang
Symmetry 2024, 16(10), 1359; https://doi.org/10.3390/sym16101359 - 13 Oct 2024
Cited by 4 | Viewed by 861
Abstract
Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB=C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation [...] Read more.
Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB=C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation AXB=C over the dual generalized commutative quaternions remains relatively insufficient. In this paper, we derive the necessary and sufficient conditions for the solvability of the dual generalized commutative quaternion matrix equation AXB=C. Furthermore, we provide the general solution expression for this matrix equation, when it is solvable. Finally, a numerical algorithm and an example are provided to confirm the reliability of the main conclusions. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
20 pages, 382 KiB  
Article
Spectral Properties of Dual Unit Gain Graphs
by Chunfeng Cui, Yong Lu, Liqun Qi and Ligong Wang
Symmetry 2024, 16(9), 1142; https://doi.org/10.3390/sym16091142 - 3 Sep 2024
Cited by 4 | Viewed by 1006
Abstract
In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer [...] Read more.
In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
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16 pages, 279 KiB  
Article
Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices
by Ruyu Tao, Ying Li, Mingcui Zhang, Xiaochen Liu and Musheng Wei
Symmetry 2024, 16(9), 1117; https://doi.org/10.3390/sym16091117 - 28 Aug 2024
Viewed by 1041
Abstract
Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation [...] Read more.
Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation AXB+CYD≈E, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation AXB+CYD≈E. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
27 pages, 355 KiB  
Article
Perturbation of Dual Group Generalized Inverse and Group Inverse
by Tianhe Jiang, Hongxing Wang and Yimin Wei
Symmetry 2024, 16(9), 1103; https://doi.org/10.3390/sym16091103 - 23 Aug 2024
Cited by 1 | Viewed by 776
Abstract
Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A^, (which is usually [...] Read more.
Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A^, (which is usually represented in the form A^=A+εA0, A and A0 are, respectively, the standard part and the infinitesimal part of A^) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0. Full article
(This article belongs to the Special Issue Exploring Symmetry in Dual Quaternion Matrices and Matrix Equations)
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