求解四元数矩阵方程A^(H)XA=B的基于矩阵半张量积的新方法  

A Real Method Based on Semi-tensor Product for Solving Quaternion Matrix Equation A^(H)XA=B

作　　者：李莹[1] 丁文旭 王栋 LI Ying;DING Wenxu;WANG Dong(School of Mathematical Sciences,Liaocheng University,Liaocheng,Shandon)

机构地区：[1]聊城大学数学科学学院,山东聊城252000

出　　处：《工程数学学报》2023年第1期159-170,共12页Chinese Journal of Engineering Mathematics

基　　金：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)。

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^(H)XA=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。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^(H)XA=B are proposed,and the corresponding algorithm is given.The e?ectiveness of the real vector representation method is veri?ed by numerical experiments.

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 

分 类 号：O151.21[理学—数学]