【作者】 张华珍；

【导师】 欧阳柏玉；

【作者基本信息】 湖南师范大学 ， 基础数学， 2008， 硕士

【摘要】 本论文研究几类约束矩阵反问题及其最佳逼近,由六章构成.在第1章,我们对约束矩阵方程反问题的历史背景与现状进行了综述.在第2章,我们运用奇异值分解给出了方程||AX-Z||²+||Y^TA-W^T||² =min存在Hermite广义Hamilton解的充要条件及其解的表达式.另外给出了矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式.在第3章,我们我们给出了线性流形S={A∈HHC^n×n|AX₁ =B₁, B₁₂X₁₁⁺ X₁₁= B₁₂, B₁₁X₁₂⁺X₁₂= B₁₁ , X₁₁^HB₁₁=B₁₂^HX₁₂},上矩阵方程||AX₂-B₂||=min的Hermite广义Hamilton解的表达式,另外,导出了在线性流形上矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式.在第4章,我们研究了线性流形S={X∈ABSR^n×n|||XY-Z| = min}上矩阵方程AXA^T=B的双反对称解.利用矩阵对的商奇异值分解,给出了这类线性流形上矩阵方程AXA^T=B存在双反对称解的充要条件及其通解表达式.另外,导出了在线性流形上矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式.在第5章,我们给出了矩阵方程（?）=min存在可对称正定解和可对称半正定解的充要条件及其通解表达式,同时解决了对称半正定化解对已知矩阵的最佳逼近问题.在第6章,我们研究了线性流形S={A∈D^-2ASR^n×n||AX-B||=min, X,B∈R^n×m}上矩阵方程f（A）=||AY-Z||=min的D反对称解,利用矩阵的奇异值分解,给出了这类线性流形上矩阵方程存在D反对称解的充要条件及其通解表达式,另外,导出了在线性流形上矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式.更多还原

【Abstract】 This thesis considers several inverse problems of constrained matrix and their best approximation. It consists of six chapters.In chapter 1,the background and present conditions are introduced and summarized for the study of inverse problems of constrained matrix and their best approximation.In chapter 2, we discuss the solvablity on the Hermite generalized Hamilton solutions of matrix equation ||AX-Z||²+||Y^TA-W^T||² =min by applying the singulau value decomposition of a matrix, obtain the necessary and sufficient conditions for the existence and the expressions for the Hermite generalized Hamilton solutions of matrix equation ||AX -Z||² +||Y^TA-W^T||² =min.In addition , in the solution set of corresponding equation,the expression of the optimal approximation solution to given matrix is derived.In chapter 3, we study the Hermite generalized Hamilton solutions of the matrix equation||AX₂-B₂|| = min on the linear manifold S = {A∈HHC^n×n|AX₁ =B₁, B₁₂X₁₁⁺ X₁₁= B₁₂, B₁₁X₁₂⁺X₁₂= B₁₁ , X₁₁^HB₁₁ =B₁₂^HX₁₂}, give the expressions for the Hermite generalized Hamilton solutions of the matrix equation||AX₂-B₂|| = min, In addition , in the solution set of corresponding equation, the expression of the optimal approximation solution to given matrix is derived.In chapter 4, we study the anti-bisymmetric solutions of matrix equation AXA^T=B on the linear manifold S = {X∈ABSR^n×n |||XY-Z|| = min}, using the quotient singula value decomposition（QSVD）of matrix pairs , obtain the necessary and sufficient conditions for the existence and the express-ions for the anti-bisymmetric solutions of matrix equation AXA^T=B . In addition , in the solution set of corresponding equation, the expression of the optimal approximation solution to given matrix is derived .In chapter 5, we disuss the necessary and sufficient conditions for the existence and the expressions for the positive definite （semidefinite） symmetrizable solutions of matrix equation （?） = min, and derive the optimal approximation semidefinite symmetrizable solutions to given matrices.In Chapter 6, we consider the D-anti-symmetric solutions of the matrix equation f（A） = ||AY-Z|| =min on the linear manifold S={A∈D^-2ASR^n×n||AX-B||=min, X,B∈R^n×m}, by applying the singulau value decomposition of a matrix, obtain the necessary and sufficient conditions for the existence and the expressions of the general Solution for the inverse problem of these matrices. For any n-by-n real matrix, it’s optimal approximation was obtained.更多还原