求解四元数矩阵方程的矩阵半张量积方法

doi:10.6040/j.issn.1671-9352.0.2020.682

摘要/Abstract

摘要： 提出四元数矩阵的一种实向量表示方法,将它应用于四元数矩阵方程的Hermitian和anti-Hermitian解的研究。通过这一实向量表示方法与矩阵半张量积结合,将四元数矩阵方程的求解问题转化成实数域中的相应问题。然后利用四元数Hermitian和anti-Hermitian 矩阵的结构特点,提取解的实向量表示中的有效信息,去除冗余,降维简化了计算的复杂度。该方法适用于具有不同约束条件的相似问题。

关键词: 四元数矩阵方程, 矩阵半张量积, 换位矩阵, 实向量表示, Hermitian(anti-Hermitian)矩阵

Abstract: A new kind of real vector representation of quaternion matrix is proposed which is applied to study the Hermitian or anti-Hermitian solution of the quaternion matrix equation. Combined this real vector representation with semi-tensor product of matrics, the problem of quaternion matrix equation is transformed into the corresponding problem in real number field. And then, by studying the structural characteristics of quaternion Hermitian matrix and anti-Hermitian matrix, the useful information in the real vector representation is extracted and the redundancy is removed, then the original problem is simplified by reducing the dimension. This method can be applied to similar problems with different constraint.

Key words: quaternion matrix equation, semi-tensor product of matrices, swap matrix, real vector representation, Hermitian(anti-Hermitian)matrix

中图分类号:

O241.6

丁文旭,李莹,王栋,赵建立. 求解四元数矩阵方程的矩阵半张量积方法[J]. 《山东大学学报(理学版)》, 2021, 56(6): 103-110.

DING Wen-xu, LI Ying, WANG Dong, ZHAO Jian-li. Solutions of the quaternion matrix equation based on semi-tensor product of matrices[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 103-110.

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