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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (5): 813-825.doi: 10.3969/j.issn.1005-3085.2022.05.010

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分裂四元数矩阵方程$AXB+CYD=E$的最小二乘$\eta$-埃尔米特解问题

张  颖,   王伟华,   魏佳宁,   张会生   

  1. 大连海事大学理学院，大连  116026
  • 出版日期:2022-10-15 发布日期:2022-12-15
  • 基金资助:
    国家自然科学基金 (61671099).

The Least Squares $\eta$-Hermitian Problems of Split Quaternion Matrix Equation $AXB+CYD=E$

ZHANG Ying,   WANG Weihua,   WEI Jianing,   ZHANG Huisheng   

  1. School of Science, Dalian Maritime University, Dalian 116026
  • Online:2022-10-15 Published:2022-12-15
  • Supported by:
    The National Natural Science Foundation of China (61671099).

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摘要：

分裂四元数矩阵方程求约束解问题在数学研究和物理应用中有重要的科学意义，针对分裂四元数矩阵的范数定义所造成的最小二乘解求解困难问题，研究了分裂四元数矩阵方程$AXB +CY D = E$的最小二乘$\eta$-埃尔米特解。首先定义分裂四元数反对合变换和$\eta$-埃尔米特矩阵，其次引入分裂四元数矩阵的Frobenius范数，通过基于分裂四元数矩阵的复表示，解决最小二乘解的求解困难问题。最后利用矩阵的Moore-Penrose广义逆以及Kronecker积，推导出分裂四元数矩阵方程的最小二乘$\eta$-埃尔米特解以及唯一极小范数解的表达式。数值实验验证了该方法的可行性。

关键词: 分裂四元数矩阵, 矩阵方程, Kronecker积, Moore-Penrose广义逆

Abstract:

The split quaternion constrained matrix equation is an important problem in mathematical research and physical applications. To overcome the difficulty in obtaining the least-squares solution of the split quaternion matrix equation, the least-squares $\eta$-Hermitian solution of the split quaternion matrix equation $AXB + CYD = E$ is considered. Firstly, anti-involution transformation and $\eta$-Hermitian matrix based on split quaternion are defined. Secondly, the Frobenius norm of a split quaternion matrix is introduced based on the complex representation of a split quaternion matrix, which dissolves the aforementioned difficulty in solving the least-squares solution. Finally, by applying the Moore-Penrose generalized inverse and Kronecker product of matrices, the least squares $\eta$-Hermitian solution and unique minimal norm solution of the split quaternion matrix equation are deduced. The feasibility of the proposed approach is verified by the numerical example.

Key words: split quaternion matrix, matrix equation, Kronecker product, Moore-Penrose generalized inverse

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