几类约束矩阵方程及其最佳逼近

【作者】 周富照；

【导师】 胡锡炎； 张磊；

【作者基本信息】 湖南大学 ， 应用数学， 2003， 博士

【摘要】 约束矩阵方程问题广泛应用于自动控制、经济、振动理论以及土木工程等。本篇博士论文系统地研究了几类约束矩阵方程问题，主要讨论的问题如下： 问题Ⅰ给定，求A∈S使AX=X(?)． 问题Ⅱ给定，集合，找A∈S使AX=B。 问题Ⅲ给定，集合，找A∈S使。 问题Ⅳ给定，找使， 这里S_E为问题Ⅰ，问题Ⅱ或问题Ⅲ的解集合。 本文的主要研究成果如下： 1．当S是所有中心对称矩阵(不必双对称)集合时，我们利用这类阵的特征性质和几何结构，获得了问题Ⅰ、Ⅱ解存在的充要条件和问题Ⅰ、Ⅱ、Ⅲ、Ⅳ解的表达式。还就S为反中心对称矩阵集合，分别对上述问题进行了讨论。 2．当S是所有对称正交对称矩阵集合时，我们首先研究了这类矩阵的几何结构和特征性质。然后分别利用特征性质、正投影阵和矩阵类通式讨论了问题Ⅰ、Ⅱ、Ⅲ及相应问题Ⅳ，获得了问题Ⅰ、Ⅱ解存在的充要条件和问题Ⅰ、Ⅱ、Ⅲ及相应问题Ⅳ解的表达式。还就相关的问题作了全面讨论，获得了许多很好结果。 3．当S是所有对称正交反对称矩阵集合时，我们讨论了这类矩阵的几何结构，获得了问题Ⅰ解存在的充要条件和问题Ⅰ、Ⅲ及相应问题Ⅳ解的表达式。 4．当S是所有反对称正交对称矩阵集合时，我们讨论了这类矩阵的几何结构，获得了问题Ⅲ和相应问题Ⅳ解的表达式。 5．当S是所有反对称正交反对称矩阵集合时，我们首先研究了这类矩阵的几何结构和特征性质，然后利用这些结构和性质获得了问题Ⅰ解存在的充要条件及问题Ⅰ、Ⅳ解的表达式。 此篇博士论文得到了国家自然科学基金的资助。 此篇博士论文由CLATEX软件打印。更多还原

【Abstract】 The constrained matrix equation problems have been widely used in control theory, economic field, vibration theory, civil engineering and so on. We will systematically study several kinds of constrained matrix equation problems in this Ph.D. Thesis, the main problems discussed are as follow:Problem I GivenX e Cn x m, A = diag(λ1, λ2,.... λm), S Rn×n. Find A ∈ S such thatAX = X. Problem II Given X,B e Rn×m, S Rn×n. Find A ∈ S such thatAX = B. Problem III Given X,B ∈ Rn×m, S Rn×n. Find A ∈ S such that||AX -B|| = min.Problem IV Given A ∈ Rn×n. Find A’ ∈ SE such that||A*-A||= min||A-~A||. A∈Sswhere SB is the solution set of Problem I or II or III The main results of this paper are as follows:1. When S is the set of all centrosymmetric matrices(nonnecessary bisymmetric), we have obtained the solvability conditions of Problem I , n and the expressions of solutions of ProblemI , II, III, IV by using the eigen-pair character and construction of this kind of matrices. Then we have also discussed the above problems respectively, when 5 is the set of all anti-centrosymmetric matrices.2. When 5 is the set of all symmetric and ortho-symmetric matrices, we have first discussed the construction and eigen-pair character of this kind of matrics. Then, by applying the eigen-pair character , the orthogonal projection and the general form of this kind of matrices, we have studied Problem I , II , in and IV, and obtained the solvability conditions of Problem I s n and the expressions of solutions of Problem I , II, III, IV. We have also solved the related problems.3. When 5 is the set of all symmetric and ortho-antisymmetric matrices, we have discussed the construction of this kind of matrices, and obtained the solvability conditions of Problem I and the expressions of solutions of Problem III and IV.4. When 5 is the set of all anti-symmetric and ortho-symmetric matrices, we have discussed the construction of this kind of matrices, and obtained the expressions of solutions of Problem III and IV.5. When 5 is the set of all anti-symmetric and ortho-antisymmetric matrices, we have first discussed the construction and eigen-pair character of this kind of matrices, then obtained the solvability conditions of Problem I and the expressions of solutions of Problem I and IV.This Ph.D.Thesis is supported by the National Natural Science Foundation of China. This Ph.D.Thesis is typeset by CLATEX .更多还原