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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 ›› 2023, Vol. 40 ›› Issue (1): 159-170.doi: 10.3969/j.issn.1005-3085.2023.01.012

• • 上一篇    

求解四元数矩阵方程$A^HXA=B$的基于矩阵半张量积的新方法

李  莹,   丁文旭,   王  栋   

  1. 聊城大学数学科学学院，山东 聊城 252000
  • 出版日期:2023-02-15 发布日期:2023-04-11
  • 基金资助:
    山东省自然科学基金(ZR2020MA053; ZR2022MA030)；聊城大学智能科学与技术强特色学科基金(319462208).

A Real Method Based on Semi-tensor Product for Solving Quaternion Matrix Equation $A^HXA=B$

LI Ying,   DING Wenxu,   WANG Dong   

  1. School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong 252000
  • Online:2023-02-15 Published:2023-04-11
  • Supported by:
    The Natural Science Foundation of Shandong Province (ZR2020MA053; ZR2022 MA030); the Discipline with Strong Characteristics of Liaocheng University-Intelligent Science and Technology (319462208).

PDF (PC)

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摘要：

四元数线性系统在控制理论和工程中有广泛的应用。利用矩阵半张量积对四元数矩阵方程进行研究，提出四元数矩阵的一种实向量表示并研究其性质。结合实向量表示与矩阵半张量积，给出四元数矩阵方程$A^HXA=B$的极小范数Hermitian解的存在条件及通解表达式，并且给出相应算法。数值实验证明了实向量表示方法的可行性。

关键词: 四元数矩阵方程, Hermitian矩阵, 矩阵半张量积, 实列(行)排, 换位矩阵

Abstract:

Quaternion linear systems have wide applications in both the control theory and engineering. Quaternion matrix equations are studied by means of semi-tensor product of matrices. A real vector representation of quaternion matrix is presented and its properties are studied. Combining the real vector representation and the semi-tensor product of matrices, the equivalent conditions for existence of Hermitian solution and the expressions of the minimal norm Hermitian solution of quaternion matrix equation $A^HXA=B$ are proposed, and the corresponding algorithm is given. The effectiveness of the real vector representation method is verified by numerical experiments.

Key words: quaternion matrix equation, Hermitian matrix, semi-tensor product of matrices, real column (row) stacking form, swap matrix

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