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Symmetry
Special Issues
The Quaternion Matrix and Its Applications
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The Quaternion Matrix and Its Applications

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 7370

Special Issue Editors

Prof. Dr. Qing-Wen Wang

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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
Dr. Zhuo-Heng He

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Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: matrix theory; quaternion algebra; numerical linear algebra
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Zhigang Jia

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Guest Editor
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
Interests: quaternion matrix theory; image processing; machine learning; scientific computing
Special Issues, Collections and Topics in MDPI journals
Dr. Guangjing Song

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Guest Editor
School of Mathematics and Information Sciences, Weifang University, Shangdong 261061, China
Interests: quaternion matrix theory; tensor theory; image processing
Dr. Yang Zhang

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Guest Editor
Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
Interests: matrix theory; tensor theory; quaternion; computer algebra

Special Issue Information

Dear Colleagues,

In recent years, quaternion matrix decomposition theory, quaternion matrix eigenvalue theory, special solutions (Hermitian, generalized Hermitian, positive definite, real part symmetric) to quaternion matrix equation or systems, to name but a few examples, have been active areas of research. In color image processing, we can encode the red, green, and blue channel pixel values on the three imaginary parts of a quaternion so that certain properties can be retained as much as possible. As a result, the quaternion matrix model can be widely used in image compression, denoising, and restoration, among numerous other applications. The real matrix representation of a quaternion matrix with generalized symmetric structure plays an important role in quaternion matrix computation.

The goal of this Special Issue is to attract original research papers on the models, theory, algorithms, and applications associated with quaternion matrices. These applications include computer vision, image and video processing, graph and network analysis, and other data-driven applications.

Prof. Dr. Qing-Wen Wang
Dr. Zhuo-Heng He
Prof. Dr. Zhigang Jia
Dr. Guangjing Song
Dr. Yang Zhang
Guest Editors

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

  • quaternion matrix
  • color image
  • singular value decomposition
  • structure preserving algorithm
  • eigenvalue of quaternion matrix
  • quaternion matrix equation

Published Papers (5 papers)

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Research

6 pages, 243 KiB  
Article
Bi-Condition of Existence for a Compatible Directed Order on an Arbitrary Field
by Niels Schwartz and YiChuan Yang
Symmetry 2023, 15(1), 215; https://doi.org/10.3390/sym15010215 - 12 Jan 2023
Viewed by 669
Abstract
One proves that a field carries a compatible directed order if and only if it has characteristic 0 and is real or has a transcendence degree of at least 1 over the field of rational numbers. Full article
(This article belongs to the Special Issue The Quaternion Matrix and Its Applications)
12 pages, 280 KiB  
Article
Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions
by Xin Liu, Kaiqi Wen and Yang Zhang
Symmetry 2023, 15(1), 40; https://doi.org/10.3390/sym15010040 - 23 Dec 2022
Viewed by 881
Abstract
We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is [...] Read more.
We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover, we obtain the explicit expressions of the least-squares reflexive (or anti-reflexive) solutions. Full article
(This article belongs to the Special Issue The Quaternion Matrix and Its Applications)
21 pages, 522 KiB  
Article
Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices
by Xueling Fan, Ying Li, Zhihong Liu and Jianli Zhao
Symmetry 2022, 14(7), 1359; https://doi.org/10.3390/sym14071359 - 01 Jul 2022
Cited by 5 | Viewed by 1405
Abstract
In this paper, we use semi-tensor product of quaternion matrices, L-representation of quaternion matrices, and GH-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices [...] Read more.
In this paper, we use semi-tensor product of quaternion matrices, L-representation of quaternion matrices, and GH-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and L-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by GH-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model. Full article
(This article belongs to the Special Issue The Quaternion Matrix and Its Applications)
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26 pages, 400 KiB  
Article
An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations
by Jing Jiang and Ning Li
Symmetry 2022, 14(4), 776; https://doi.org/10.3390/sym14040776 - 08 Apr 2022
Cited by 2 | Viewed by 1239
Abstract
In the present paper, an iterative algorithm is proposed for solving the generalized (P,Q)-reflexive solution group of a system of quaternion matrix equations [...] Read more.
In the present paper, an iterative algorithm is proposed for solving the generalized (P,Q)-reflexive solution group of a system of quaternion matrix equations ∑l=1M(AlsXlBls+ClsXl˜Dls)=Fs,s=1,2,…,N. A generalized (P,Q)-reflexive solution group, as well as the least Frobenius norm generalized (P,Q)-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized (P,Q)-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized (P,Q)-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. Full article
(This article belongs to the Special Issue The Quaternion Matrix and Its Applications)
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19 pages, 311 KiB  
Article
Three Symmetrical Systems of Coupled Sylvester-like Quaternion Matrix Equations
by Mahmoud Saad Mehany and Qing-Wen Wang
Symmetry 2022, 14(3), 550; https://doi.org/10.3390/sym14030550 - 08 Mar 2022
Cited by 23 | Viewed by 1828
Abstract
The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are [...] Read more.
The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper. Full article
(This article belongs to the Special Issue The Quaternion Matrix and Its Applications)
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