四元数矩阵方程最小二乘Toeplitz解的半张量积方法

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (6): 154-159.

四元数矩阵方程最小二乘Toeplitz解的半张量积方法

闫立梅^*1, 赵琳琳¹, 丁文旭², 李莹², 范洪彪¹

1.德州学院数学与大数据学院, 山东德州 253000;
2.聊城大学数学科学学院, 山东聊城 252000

收稿日期:2022-05-30 出版日期:2023-12-28 发布日期:2024-01-05
通讯作者: 闫立梅(1969-),女,山东德州人,教授.Email:yanlimei9898@163.com
基金资助:
山东省自然科学基金(ZR2020MA053)

Semi-tensor product method for solving least square Toeplitz solutions of quaternion matrix equation

YAN Li-mei¹, ZHAO Lin-lin¹, DING Wen-xu², LI Ying², FAN Hong-biao¹

1. School of Mathematics and Big Data, Dezhou University, Dezhou 253000, China;
2. School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China

Received:2022-05-30 Online:2023-12-28 Published:2024-01-05

摘要/Abstract

摘要： 研究了四元数矩阵方程$\sum_{i=1}^{k} \boldsymbol{A}_{i} \boldsymbol{X}_{i} \boldsymbol{B}_{i}=\boldsymbol{C}$的最小二乘Toeplitz解和Hermitian Toeplitz解的问题.联合使用四元数矩阵的实向量表示方法和矩阵的半张量积方法,将所研究的问题转化为实矩阵方程.根据Toeplitz矩阵以及Hermitian Toeplitz矩阵的结构特征,提取了矩阵中的有效元素,构造了新的解向量,降低了所研究问题的复杂度.得到了方程存在Toeplitz解和Hermitian Toeplitz解的条件,并给出Toeplitz解和Hermitian Toeplitz解的一般形式.通过数值算例说明了方法的精度和算法的可行性.

关键词: 四元数矩阵方程, 矩阵半张量积, 最小二乘Toeplitz解, 最小二乘Hermitian Toeplitz解

Abstract: Least square Toeplitz solutions and Hermitian Toeplitz solutions of the quaternion matrix equation $\sum_{i=1}^{k} \boldsymbol{A}_{i} \boldsymbol{X}_{i} \boldsymbol{B}_{i}=\boldsymbol{C}$ are studied. Utilizing real vector representation of the quaternion matrix and semi-sensor product theory, the quaternion matrix equation is transformed into its equivalent real matrix equation. Considering the structural characteristics of the Toeplitz matrix and Hermitian Toeplitz matrix, independent elements of the solution matrix are extracted to reconstruct a new solution vector, thus the computational complexity of the problem is reduced. The existing conditions of Toeplitz solutions and Hermitian Toeplitz solutions of the equation are obtained, and the general solutions of the equation are given. Finally, a numerical example is given to demonstrate the precision degree and effectiveness of the algorithm.

Key words: quaternion matrix equation, semi-tensor product of matrices, least square Toeplitz solution, least square Hermitian Toeplitz solution

中图分类号:

O241.6

闫立梅, 赵琳琳, 丁文旭, 李莹, 范洪彪. 四元数矩阵方程最小二乘Toeplitz解的半张量积方法[J]. 兰州理工大学学报, 2023, 49(6): 154-159.

YAN Li-mei, ZHAO Lin-lin, DING Wen-xu, LI Ying, FAN Hong-biao. Semi-tensor product method for solving least square Toeplitz solutions of quaternion matrix equation[J]. Journal of Lanzhou University of Technology, 2023, 49(6): 154-159.

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