兰州大学机构库 >数学与统计学院
两类约束矩阵方程的解及最佳逼近问题
Alternative TitleThe Solution for Two Kinds Constrained Matrix Equation Problems and Associated Optimal Approximation
高红桃
Thesis Advisor尤传华
2009-05-30
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword约束矩阵方程 (R,S) 对称矩阵 (R,S) 反对称矩阵 Frobenius范数 广义奇异值分解 迭代算法 极小范数解 最佳逼近
Abstract在给定特殊矩阵集合中, 求矩阵方程的解, 即为约束矩阵方程问题. 在本文中, 我们介绍了(R,S) 对称矩阵, (R,S) 反对称矩阵的概念及结构. 在这些特殊矩阵集合中, 我们运用矩阵广义奇异值分解, 得到了矩阵方程 AXB = C 的(R,S) 反对称解;通过构造一种有效的迭代算法, 得到了矩阵方程 AXB + CYD = E 的(R,S) 对称解, 同时考虑了相应的最佳逼近问题. 根据(R,S) 反对称矩阵的性质,我们得到了阵方程 AXB=C 有(R,S) 反对称解的充要条件, 并给出了一般解的表达式, 在此基础上, 对任意给定的矩阵X*<上标!>属于C m*n<上标!>, 我们给出了最佳唯一逼近解X^ <上标!>属于SE<下标!> 及其表达式. 针对求解AXB + CYD=E 迭代算法, 我们证明了, 在不考虑机器误差的情况下,对任意初始迭代矩阵[X1<下标!>,Y1<下标!>],矩阵方程的解 [X,Y] 可以经过有限步迭代得到, 且矩阵方程AXB + CYD=E 的相容性能够自动判断. 如果取特殊形式的[X1<下标!>,Y1<下标!>](比如X1<下标!>=0, Y1<下标!>=0), 则由迭代算法得到的解是矩阵方程的极小范数解. 另外当上述方程相容时, 在这些矩阵的解 集中, 对于任意给定矩阵对[X*<上标!>,Y*<上标!>] 的最佳逼近解[ X^,Y^], 可以通过求解新的约束矩阵方程AX~B + C Y~D =E~ 极小范数解[X~*<上标!> ; Y~*<上标!> ] 得到(利用上述迭代解法), 其中X~<上标!>= X-X*<上标!>,Y~ = Y-Y*<上标!>; E~ <上标!>= E-AX~<上标!>B-CY~<上标!>D, 从而X^= X~*<上标!>+X*<上标!>, Y^= Y~*<上标!>+Y*<上标!>,对于迭代算法, 我们给出数值例子, 说明该算法的可行性和有效性.
Other AbstractTo find solution of matrix equation in some given matrix sets is so-called the constrained matrix problem. In this paper, we first introduce the concepts and structures of (R,S)-symmetric-matrices, (R,S)-skew-symmetric matrices. Whereafter, we derive the (R,S)-skew-symmetric solution for the matrix equation AXB=C by using generalized singular-value decomposition; We get the (R,S)-symmetric solution for the matrix equation AXB + CYD =E by establishing an efficient algorithm. Meanwhile,the optimal unique approximation is considered.According to the properties of (R,S)-skew-symmetric matrices, we derive the necessary and sufficient conditions and expression for (R; S)-skew-symmetric solution of matrix equation AXB =C . Moreover, for a arbitrary given matrix X*<上标!>belongs to C m*n<上标!>,the optimal unique approximation X^ <上标!> belongs to SE<下标!> and its expression is provided. For the iterative method of the matrix equation AXB + CYD =E , we prove that, for arbitrary initial iterative matrix pair [X1<下标!>,Y1<下标!>], the solution of matrix equation can be obtained within finite iterative steps in the absence of roundoff errors and the solvability of the matrix equation can be determined automatically in the iterative process. Especially,if let X1<下标!> = 0, Y1<下标!> = 0 or X1<下标!>,Y1<下标!> have some particular form, the solution obtained by the iterative algorithm is the least Frobenius norm solution. In addition, by the iterative algorithm, we can represent the associated optimal approximation solutionpair [ X^,Y^ ] for any arbitrary given matrix pair [X*<上标!>,Y*<上标!>]which can be derived by the least-norm solution [ X~*<上标!> ; Y~*<上标!> ]of the new matrix equation AX~B + C Y~D =E~, where X~<上标!>= X-X*<上标!>,Y~ = Y-Y*<上标!>; E~ <上标!>= E-AX~<上标!>B-CY~<上标!>DFinally, we give associated numerical examples, which illustrate the efficiency of the iterative methods.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224880
Collection数学与统计学院
Recommended Citation
GB/T 7714
高红桃. 两类约束矩阵方程的解及最佳逼近问题[D]. 兰州. 兰州大学,2009.
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