广西师范大学学报（自然科学版） ›› 2022, Vol. 40 ›› Issue (4): 136-144.doi: 10.16088/j.issn.1001-6600.2021092301

• 研究论文 • 上一篇    下一篇

矩阵方程的秩约束最小二乘对称半正定解及其最佳逼近

喻思婷, 彭靖静*, 彭振赟   

  1. 桂林电子科技大学数学与计算科学学院,广西桂林 541004
  • 发布日期:2022-08-05
  • 通讯作者: 彭靖静(1987—),男,湖南邵阳人,桂林电子科技大学副研究员,博士。E-mail: jjpeng2012@163.com
  • 基金资助:
    国家自然科学基金(11961012); 广西自然科学基金(2018GXNSFBA281192);广西科技基地和人才专项项目(AD20297063)

Rank Constraint Least Square Symmetric Semidefinite Solutions and Its Optimal Approximation of the Matrix Equation

YU Siting, PENG Jingjing*, PENG Zhenyun   

  1. College of Mathematics and Computer Science, Guilin University of Electronic Technology, Guilin Guangxi 541004,China
  • Published:2022-08-05

PDF (PC)

169

摘要： 基于矩阵的奇异值分解和对称矩阵谱分解,给出矩阵方程AX=B有秩约束最小二乘对称半正定解及其最佳逼近解的充分必要条件及有解时解的一般表达式;给出求解最佳逼近解的计算步骤;用数值例子说明结果的正确性。

关键词: 秩约束矩阵, 矩阵方程, 对称半正定矩阵, 最小二乘解, 最佳逼近

Abstract: Based on the singular value decomposition and spectral decomposition of matrices, the necessary and sufficient conditions for the existence of solutions to rank constraint least square symmetric semidefinite solutions and itsoptimal approximation solution of the matrix equation AX=B are established and, if the solutions exist, the general expression of the solutions are proposed. The computational procedures of the optimal approximation solution, and the numerical examplesshowing the correctness of the theoretical results are given.

Key words: rank constraint matrix, matrix equation, symmetric semidefinite matrices, least square solution, optimal approximation

中图分类号: 

  • O151.21
[1] BJERHAMMAR A. Rectangular reciprocal matriceswith special reference to geodetic calculations[J]. Bulletiin Géodésique(1946-1975), 1951, 20(1): 188-220.
[2]廖安平. 线性流形上矩阵方程AX=B的一类反问题及数值解法[J]. 计算数学,1988, 20(4): 371-376.
[3]HIGHAM N J. The symmetric procrustes problem[J]. Bit Numerical Mathematics, 1988, 28(1): 133-143.
[4]ALLWRIGHT J C. Positive semidefinite matrices: characterization via conical hulls and least-squares solution of a matrix equation[J]. SIAM Journal on Control Optimization, 1988, 26(3): 537-556.
[5]HUA D. On the symmetric solutions of linear matrix equations[J]. Linear Algebra and Its Applications, 1990, 131: 1-7.
[6]张磊,谢冬秀,胡锡炎. 线性流形上双对称阵逆特征值问题[J]. 计算数学,2000, 22(2): 129-138.
[7]XIE D X, SHENG Y P. Inverse eigenproblem of anti-symmetric and persymmetric matrices and its approximation[J]. Inverse Problems, 2003, 19(1): 217-225.
[8]彭振赟. 几类矩阵扩充问题和几类矩阵方程问题[D]. 长沙:湖南大学,2003.
[9]LI F Y,PENG J J,PENG Z Y. The least squares stochatic solutions of the matrix equation AX=B[J]. Journal of Algebra Number Theory:Advances and Applications, 2012, 8(1/2):19-39.
[10]徐安豹,彭振赟. 矩阵方程AX＝B 的范数约束最小二乘解[J]. 桂林电子科技大学学报, 2013, 33(1): 70-73.
[11]彭金凤,彭振赟,崔学莲. 矩阵方程AX＝B的D对称半正定最小二乘解及其最佳逼近[J]. 桂林电子科技大学学报,2018, 38(2): 162-166.
[12]李媛,谢冬秀. 矩阵方程AX=B的可双对称化解及其最佳逼近[J]. 北京信息科技大学学报(自然科学版),2020, 35(1): 49-53.
[13]张四保,邓勇. 主理想环上矩阵方程AX=B的对称解[J]. 科学技术与工程,2020, 20(25): 10133-10137.
[14]ALBERT A. Condition for positive and nonnegative definite in terms of pseudoinverse[J]. SIAM Journal on Applied Mathematics, 1969, 17(2): 434-440.
[15]PENG J J, WANG Q W, PENG Z Y, et al. Solution of symmetric positive semidefinite procrustes problem[J]. Electronic Journal of Linear Algebra, 2019, 35: 543-554.
[16]WEI M, WANG Q. On rank-constrained-Hermitian nonnegative-denite least squares solutions to the matrix equation AXAH=B[J]. International Journal Computer-Mathematics, 2007, 84(6): 945-952.
[17]ZHANG X, CHENG M Y. The rank-constrained Hermitian nonnegative definite and positive-definite solutions to the matrix equation AXA*=B[J]. Linear Algebra and Its Applications, 2003,370: 163-174.
[18]XIAO Q F, HU X Y, ZHANG L. The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximetion[J]. Electronic Journal of Linear Algebra, 2009, 18: 264-273.
[1] 武康康, 朱旭飞, 陆叶, 周鹏, 董翠, 戴沁璇, 周闰昌. 基于最小二乘法的LS-FIR滤波器[J]. 广西师范大学学报（自然科学版）, 2021, 39(5): 89-99.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 广西师范大学学报（自然科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发