矩阵方程AXB = C的轮换极小范数最小二乘解
Least Squares Circulant Solution of the Matrix Eqaution AXB = C with the Least Norm
DOI: 10.12677/PM.2022.128149, PDF, HTML, 下载: 259  浏览: 381  科研立项经费支持
作者: 曹煜喆, 袁仕芳*：五邑大学数学与计算科学学院，广东江门
关键词: 轮换矩阵；极小范数解；最小二乘解；Moore-Penrose广义逆；Kronecker 积；Circulant Matrix； Least Norm Solution； Least Squares Solution； Moore-Penrose Inverse； The Kronecker Product
摘要: 循环矩阵有悠久的历史并且在众多科学领域得到了广泛的应用。矩阵方程AXB=C在特定集合类的求解和最小化问题在工程等领域有重要的应用。本文通过矩阵的Kronecker积和Moore-Penrose广义逆得到了矩阵方程AXB=C有轮换解的充要条件和解的表达式。在没有轮换解时，给出了方程的轮换极小范数最小二乘解。在论文末节，给出方程求解的数值算法与数值例子。
Abstract: Circulant matrices have been around for a long time and have been extensively used inmany scientific areas. The problem of solving and minimizing the matrix AXB = C in a specific set class has important applications in engineering and other related fields. In this paper, by using Kronecker product and Moore-Penrose generalized inverse of the matrices, the necessary and suficient conditions for AXB = C having circulant solution are obtained. We derive the expression of the least squares circulant solution of the matrix equation AXB = C with the least norm when there is no circulant solution. In the last section, the numerical algorithm and numerical examples are also given.
文章引用：曹煜喆, 袁仕芳. 矩阵方程AXB = C的轮换极小范数最小二乘解[J]. 理论数学, 2022, 12(8): 1360-1369. https://doi.org/10.12677/PM.2022.128149

参考文献

[1] Hu, X.Y. and Deng, Y.B. (2003) On the Solutions Optimal Approximation of the Equation ATXB = C over ASRmxm. Numerical Mathematics, 5, 59-62.
[2] Liao, A.P., Bai, Z.Z. and Lei, Y. (2006) Best Approximate Solution of Matrix Equation AXB+ CXD = E. SIAM Journal on Matrix Analysis and Applications, 27, 675-688.
https://doi.org/10.1137/040615791
[3] 袁仕芳，廖安平，雷渊.矩阵方程AXB+CYD=E的对称极小范数最小二乘解[J].计算数学，2007, 29(2): 203-216.
[4] Yuan, S. and Liao, A. (2014) Least Squares Hermitian Solution of the Complex Matrix Equa- tion AXB + CXD = E with the Least Norm. Journal of the Franklin Institute, 351, 4978-4997.
https://doi.org/10.1016/j.jfranklin.2014.08.003
[5] Liu, Z., Zhou, Y., Zhang, Y., et al. (2019) Some Remarks on Jacobi and Gauss-Seidel-Type Iteration Methods for the Matrix Equation AXB = C. Applied Mathematics and Computation, 354, 305-307.
https://doi.org/10.1016/j.amc.2019.02.014
[6] Dehghan, M. and Shirilord, A. (2019) A Generalized Modified Hermitian and Skew-Hermitian Splitting (GMHSS) Method for Solving Complex Sylvester Matrix Equation. Applied Mathe- matics and Computation, 348, 632-651.
https://doi.org/10.1016/j.amc.2018.11.064
[7] Wu, N.C., Liu, C.Z. and Zuo, Q. (2022) On the Kaczmarz Methods Based on Relaxed Greedy Selection for Solving Matrix Equation AXB = C. Journal of Computational and Applied Math- ematics, 413, Article ID: 114374.
https://doi.org/10.1016/j.cam.2022.114374
[8] Safarzadeh, M., Sadeghi Goughery, H. and Salemi, A. (2022) Global-DGMRES Method for Matrix Equation AXB = C. International Journal of Computer Mathematics, 99, 1005-1021.
https://doi.org/10.1080/00207160.2021.1942459
[9] 陈景良，陈向辉.特殊矩阵[M]. 北京：清华大学出版社, 2001.
[10] Davis, P.J. (1994) Circulant Matrices. AMS Chelsea Publishing, New York.
[11] Arup, B. and Koushik, S. (2018) Random Circulant Matrices. CRC Press, Boca Raton, FL.
[12] 戴华.矩阵[M]. 北京: 科技出版社, 2001.

为你推荐