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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 ›› 2025, Vol. 42 ›› Issue (3): 411-424.doi: 10.3969/j.issn.1005-3085.2025.03.002doi: 32411.14.1005-3085.2025.03.002

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一类耦合Sylvester四元数矩阵方程组的Cramer法则

蔡小敏,  柯艺芬   

  1. 福建师范大学数学与统计学院，福州  350117
  • 收稿日期:2022-08-12 接受日期:2023-03-29 出版日期:2025-06-15 发布日期:2025-06-15
  • 通讯作者: 柯艺芬 E-mail: keyifen@fjnu.edu.cn
  • 基金资助:
    福建省自然科学基金 (2024J01478).

Cramer's Rule for a Class of Coupled Sylvester Quaternion Matrix Equations

CAI Xiaomin,   KE Yifen   

  1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117
  • Received:2022-08-12 Accepted:2023-03-29 Online:2025-06-15 Published:2025-06-15
  • Contact: Y. Ke. E-mail address: keyifen@fjnu.edu.cn
  • Supported by:
    The Natural Science Foundation of Fujian Province (2024J01478).

摘要：

四元数矩阵方程在信号处理、图像处理、控制系统等领域具有广泛的应用。利用四元数矩阵的实表示、行列式及其逆矩阵的定义和性质，并根据四元数矩阵的Kronecker积，将一类耦合Sylvester四元数矩阵方程组转化为等价的实线性方程，研究该方程组有唯一解时的Cramer法则。最后利用四元数与其实表示之间的同态关系，推导出了耦合Sylvester四元数矩阵方程组的Cramer法则。通过对该方法的分析，证明了其在理论上的有效性和可行性。数值算例表明所提出的理论方法具有较好的计算性能和实用性。

关键词: 四元数, 耦合Sylvester矩阵方程组, Cramer法则

Abstract:

Quaternion matrix equations have widespread applications in signal processing, image processing, control systems, etc. By using the real representation, determinant, inverse matrix definition and properties of quaternion matrices, and based on the Kronecker product of quaternion matrices, a class of coupled Sylvester quaternion matrix equations is transformed into equivalent real linear equations. The Cramer's rule for the solution of the equations with a unique solution is studied. Finally, the Cramer's rule for the coupled Sylvester quaternion matrix equations is derived by using the isomorphic relationship between quaternions and their real representations. Through the analysis of the method, the theoretical validity and feasibility are proved. Numerical examples demonstrate that the proposed theoretical method has good computational performance and practicality.

Key words: quaternion, coupled Sylvester matrix equations, Cramer's rule

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