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Keywords = skew bisymmetric matrix

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14 pages, 339 KB  
Article
An Efficient Method for Split Quaternion Matrix Equation XAf(X)B = C
by Shufang Yue, Ying Li, Anli Wei and Jianli Zhao
Symmetry 2022, 14(6), 1158; https://doi.org/10.3390/sym14061158 - 4 Jun 2022
Cited by 5 | Viewed by 2300
Abstract
In this paper, we consider the split quaternion matrix equation XAf(X)B=C, f(X){X,XH,XiH,XjHXkH} [...] Read more.
In this paper, we consider the split quaternion matrix equation XAf(X)B=C, f(X){X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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14 pages, 317 KB  
Article
A New Method of Solving Special Solutions of Quaternion Generalized Lyapunov Matrix Equation
by Zhihong Liu, Ying Li, Xueling Fan and Wenxv Ding
Symmetry 2022, 14(6), 1120; https://doi.org/10.3390/sym14061120 - 29 May 2022
Cited by 2 | Viewed by 2157
Abstract
In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed [...] Read more.
In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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