导　　师： 张忠志;雷秀仁

学科专业： 070102

授予学位： 硕士

作　　者： ;

机构地区： 华南理工大学

摘　　要： 矩阵逆特征值问题就是根据给定的谱数据构造矩阵,其中给定的谱数据结构可以是全部或是部分关于特征值或特征向量的信息,主要应用于控制设计、地球物理学、分子光谱学、粒子物理学、结构分析等领域.约束矩阵问题则是在满足约束条件的矩阵集合中求矩阵方程的解的问题,其广泛用于结构设计、系统识别、结构动力学、自动控制、振动理论、计算物理学、非线性规划以及土木工程等领域.埃尔米特自反矩阵作为一个具有特殊性质的矩阵,其逆特征值问题和约束矩阵方程问题在实际中也具有广泛的应用.为此,本篇论文主要讨论埃尔米特自反矩阵的逆特征值问题以及相关约束矩阵问题,具体数学描述如下： 问题Ⅰ给定X,Y,∈Cn×m,Λ=diag/(μ1,…,μm/),Γ=/(τ1,…,τm/)∈Rm×m,求A∈HCrn×n/(P/),使得AX=XΛ,YH A=ΓYH 问题Ⅱ给定X∈Cn×m,Λ=diag/(μ1,…,μm/),求A,B∈HCrn×n/(P/),使得AX=BXA 问题Ⅲ给定A,B,C∈Cn×m,求X,Y∈HCrn×n/(P/),使得‖AX+BY-C‖=min 问题Ⅳ给定X,B∈Cn×m,求A∈HCrn×n/(P/),使得这里S是一个线性流形. 问题V对任意给定的矩阵A*,B*∈Cn×n,求A∈T,B∈T,使得或是这里T是上述Ⅰ,Ⅱ,Ⅲ,Ⅳ四个问题的解集合,HCrn×n/(P/)为埃尔米特自反矩阵的集合. 本文的主要研究的内容为： /(1/)利用埃尔米特自反矩阵的特征值和特征向量的性质,给出了其逆特征值问题的数学描述,并研究了埃尔米特自反矩阵集合的性质和结构,推导了埃尔米特自反矩阵集合的表示定理. /(2/)利用埃尔米特自反矩阵的表示定理和矩阵的奇异值分解,分别给出问题Ⅰ,Ⅳ可解的充分必要条件及可行解的一般表达式.同时,利用矩阵的表示定理和矩阵的拉直方法,分别得到了问题Ⅱ、Ⅲ的解的一般表达式. /(3/)在/(2/)研究的的基础上,进一步研究了上述三个问题在可解条件不满足时的最小二乘解问题,得到其最小二乘解的一般表达式. /(4/)对应问题Ⅴ,分别证明了上述三个问题的最佳逼近问题解得的存在性和惟一性,推导了其最佳逼近解的表达式. /(5/)最后给出对应上述各个最佳逼近问题的算法和数值算例,以验证其结果的正确性. 本篇论文得到了国家自然科学基金的资助. A matrix inverse eigenvalue problem concerns the reconstruction of a matrix from prescribed spectral date. The spectral data involved may consist of the complete or only partial information of eigenvalue or eigenvectors. The inverse eigenvalue problem is mainly used in control design, geophysics, molecular spectroscopy, particle physics, structure analysis and so on. The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set, which is applied in structural design, system identification, automatic control, vibration theory, computational physics, nonlinear programming, and civil engineering and so on. The inverse eigenvalue and matrix equation problem of Hermitian reflexive matrix is widely used in practice due to the special nature. So, in this paper, the inverse eigenvalue problem and the constraint matrix equation problem are considered, which mathematical description is as following, ProblemⅠ. Given X, Y∈Cn×m,Λ=diag /(μ1,…,μm/),Γ=/(τ1,…,τm/)∈Rm×m, find A∈HCrn×n/(P/),such that AX=XA, YH A=ΓYH ProblemⅡ. Given X∈Cn×m,Λ=diag /(μ1,…,μm/)∈Cm×m, find A, B∈HCrn×n /(P/),such that AX=BXΛ ProblemⅢ. Given A,B,C∈Cn×m, find X, Y∈HCrn×n /(P/), such that‖AX+BY-C‖=min ProblemⅣ. Given X, B∈Cn×m, find A∈HCrn×n /(P/), such that Where the S is a linear manifold. ProblemⅤ. For any given matrices A*, B*∈Cn×n,find A∈T,B∈T, such that Or Where the T is the solutions of the above problemⅠ,Ⅱ,ⅢandⅣrepectively. The main research content of this paper are as following: /(1/) By using the property of the eigenvalue and the eigenvector of the Hermitian matrix, the mathematical description on inverse eigenvalue problem is presented. The property and structure of the hermitian reflexive matrix are considered, and the denotative theorem of the hermitian reflexive matrix is derived. /(2/) By using the denotative theorem which is obtained above of hermitian reflexive matrix and the SVD decompounding method, the sufficient and necessary conditions and the expression of the general solutions for problemⅠandⅣare obtained. At the same time, by using the straightened matrix, the expressions of the general solutions of the problemⅡandⅢare obtained respectively. /(3/) Based on the 2, furthermore, the least-squares solutions of the above problem are considered, and the general expression of the solutions are presented. /(4/) Corresponding the problemⅤ, the existence and uniqueness of the optimal approximation solutions are proved respectively and the expressions of the best optimal approximation solutions are derived. /(5/) Finally, in order to prove the correctness of the method and the solutions, the numerical algorithms and the numerical example for each problem are given respectively. The dissertation has gained the support from the National Natural Science Foundation of China.

关 键 词： 埃尔米特自反矩阵 左右逆特征值 广义逆特征值 矩阵方程 线性流形 最小二乘解 最佳逼近

分 类 号： [O241.6]

领　　域： [理学] [理学]