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Search Results (3)

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Keywords = Sylvester-conjugate matrix equation

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11 pages, 292 KB  
Article
A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations
by Ryszard Mazurek
Symmetry 2024, 16(2), 246; https://doi.org/10.3390/sym16020246 - 17 Feb 2024
Cited by 1 | Viewed by 2206
Abstract
Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector [...] Read more.
Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects. Full article
(This article belongs to the Section B: Mathematics)
28 pages, 549 KB  
Article
The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations
by Xiaowen Wu, Zhengge Huang, Jingjing Cui and Yanping Long
Axioms 2023, 12(11), 1062; https://doi.org/10.3390/axioms12111062 - 20 Nov 2023
Cited by 2 | Viewed by 1806
Abstract
By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the [...] Read more.
By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm. Full article
(This article belongs to the Section Mathematical Analysis)
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18 pages, 668 KB  
Article
Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation
by Kanjanaporn Tansri and Pattrawut Chansangiam
Symmetry 2022, 14(9), 1868; https://doi.org/10.3390/sym14091868 - 7 Sep 2022
Cited by 9 | Viewed by 3029
Abstract
We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation [...] Read more.
We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods. Full article
(This article belongs to the Section B: Mathematics)
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