几类特殊线性约束矩阵方程问题及其最佳逼近问题
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摘要
约束矩阵方程问题是指在满足一定条件的矩阵集合中求给定的矩阵方程解的问题.约束矩阵集合不同,求解的矩阵方程类型不同,均构成不同的问题.随着科学与工程技术的不断发展,新的约束矩阵方程问题不断地提出,特别在控制论,信息论,振动理论,系统识别,结构动力模型修正和自动系统模拟等尤为显著,其研究所取得的成果,有直接的重要的应用背景。本篇博士论文主要是对逆特征值问题(AX=X∧)以及对矩阵方程AX=B的等式问题(即线性约束问题)和最小二乘问题进行了研究,主要工作如下:
1.系统地研究了中心主子阵约束下几类实矩阵的约束矩阵方程问题,内容包括中心主子阵约束下中心对称矩阵的逆特征值问题,线性约束问题,最小二乘问题,左右逆特征值问题(AX=X∧,Y~TA=ΓY~T),方程组(AX=Z,Y~TA=W~T)的最小二乘问题;中心主子阵约束下双对称矩阵的逆特征值问题,线性约束问题,最小二乘问题;中心主子阵约束下对称次反对称矩阵,反对称次对称矩阵,双反对称矩阵的最小二乘问题。通过分析这几类矩阵与其各自中心主子阵的特殊性质,得到了中心主子阵具有和原矩阵相同的结构,从而巧妙地将上述几类问题转化为阶数减半的几类子矩阵约束下的(左右)逆特征值问题,线性约束问题,或(方程组)最小二乘问题。这是解决问题的关键,是这部分区别于其他子矩阵约束问题的特色所在。在此基础上,彻底地解决了中心主子阵约束下的矩阵方程问题,得到了问题有解的充要条件,通解表达式,相应问题的最佳逼近解以及数值算例。
2.研究了线性流形上复对称矩阵,复双对称矩阵的最小二乘问题,利用复矩阵的奇异值分解,广义逆等得到了问题通解的表达式以及最佳逼近解。讨论了Hermite正定矩阵的最小二乘问题,通过对Hermite正定矩阵分块形式的分析,得到了矩阵存在最小二乘解及其最佳逼近解的充要条件,给出了最小二乘解和最佳逼近解的表达式,并提供了求最佳逼近解的算法和数值算例。
3.研究了Hermite自反矩阵的逆特征值问题、线性约束问题和Hermite反自反矩阵的逆特征值问题、线性约束问题。通过分析这两类矩阵特征向量的性质,巧妙合理地给出了逆特征值问题的数学描述。接着定义一种新的内积,利用Hermite(反)自反矩阵特有的结构,以及它们与对称向量,反对称向量的关系,获得了问题可解的充要条件,通解表达式,最佳逼近解以及数值算例。
4.讨论了行(列)对称矩阵,行(列)反对称矩阵,行(列)延拓矩阵的性质。研究了矩阵方程X~HAX=B的行(列)反对称问题,行(列)延拓矩阵的最小二乘问题,得到了问题有解的充要条件,通解表达式以及最佳逼近解。研究了求矩阵方程AXB=C的行(列)对称解或行(列)对称最小二乘解的两类迭代法,并对其中的一类迭代法进行了收敛性分析。最后证明了只需对算法稍加修改,则均可求得相应问题的最佳逼近解。
此博士论文得到了国家自然科学基金(10571047)和高等学校博士学科点专项科研基金(20060532014)的资助。
此博士论文用L~AT_EX2_ε软件打印。
The constrained matrix equation problem is to find solution to a matrix equation in a constrained matrix set.The different constrained condition,or the different matrix equation makes a different constrained matrix equation problem. More and more different constrained matrix equation problems arise out of the developments in science and engineering,especially in control theory,information theory,vibration theory,system identification,structural dynamics model updating problem,and mechanical system simulation.Thus the research results on these problems have useful applications.This dissertation considers inverse eigenvalue problem(AX=X∧),linear constraint problem and least squares problem for matrix equation AX=B,which are:
1.The research studies some real matrix equations problems under central principal submatrices constraint,including inverse eigenvalue problem,linear constraint problem,least squares problem,right and left inverse eigenvalues problem (i.e.AX=X∧,Y~TA=ΓY~T),and least squares problem of matrix equations (AX=Z,Y~TA=W~T) for centrosymmetric matrices under central principal submatrices constraint;inverse eigenvalue problem,linear constraint problem,and least squares problem for bisymmetric matrices under central principal submatrices constraint;least squares problems for skew symmetric and persymmetric matrices, symmetric and skew persymmetric matrices,and antisymmetric and skew persymmetric matrices under central principal submatrices constraint.By analysis of special properties and structure of these matrices and themselves central principal submatrices,the paper obtains that the submatrices having the same symmetric properties and structure as the given matrices,and converts these problems to (right and left) inverse eigenvalue problems,linear constraint problems,(matrix equations) least squares problems of half-sized real matrices under submatrices constraint. This simplifies and is crucial to solve the problems,and is a special feature of this part different from other papers about submatrix constraint problems.Base on these,it derives necessary and sufficient conditions for the solvability,representation of the general solutions,corresponding optimal approximation solutions and some numerical examples.
2.The research studies least squares problems for complex symmetric matrices and complex bisymmetric matrices on linear manifold,and obtains representation of general solutions and the corresponding optimal approximation solutions by using the singular value decomposition and Moore-Penrose generalized inverse. Moreover,the paper discusses least squares problem of Hermitian positive definite matrices,induces necessary and sufficient conditions and representation of least squares solutions and the optimal approximation solution by analysis of its block form,and gains a numerical method and a numerical experiment for optimal approximation solution.
3.The research studies inverse eigenvalue problems,linear constraint problems for Hermitian reflexive matrices and Hermitian skew reflexive matrices.It presents mathematical description of inverse eigenvalue problems subtly and reasonably by analysis of eigenvectors of these two kinds matrices.By using a new inner product,special structure of Hermitian(skew) reflexive matrices,and the relationship between them and reflexive vectors or anti-reflexive vectors,this paper obtains necessary and sufficient conditions for the solvability,representation of the general solutions,and corresponding optimal approximation solutions.
4.The research studies the properties of row(column) symmetric matrices, row(column) skew symmetric matrices and row(column) extended matrices,and considers linear constraint problem of matrix equations X~HAX=B for row(column) skew symmetric matrices,least squares problem for row(column) extended matrices.It derives necessary and sufficient conditions for the solvability,representation of the general solutions and corresponding optimal approximation solutions. This dissertation also puts forward two kinds of iteration methods for achieving the general solutions or least squares solutions to matrix equation AXB=C for row(column) symmetric matrices,and obtains the convergence rate of one iteration method of them.Moreover,the related optimal approximation solutions can be gained with the method which only needs to be made slight changes.
This dissertation is supported by the National Natural Science Foundation of China(10571047) and Doctorate Foundation of the Ministry of Education of China(20060532014).
This dissertation is typeset by software L~AT_EX2_ε.
1.系统地研究了中心主子阵约束下几类实矩阵的约束矩阵方程问题,内容包括中心主子阵约束下中心对称矩阵的逆特征值问题,线性约束问题,最小二乘问题,左右逆特征值问题(AX=X∧,Y~TA=ΓY~T),方程组(AX=Z,Y~TA=W~T)的最小二乘问题;中心主子阵约束下双对称矩阵的逆特征值问题,线性约束问题,最小二乘问题;中心主子阵约束下对称次反对称矩阵,反对称次对称矩阵,双反对称矩阵的最小二乘问题。通过分析这几类矩阵与其各自中心主子阵的特殊性质,得到了中心主子阵具有和原矩阵相同的结构,从而巧妙地将上述几类问题转化为阶数减半的几类子矩阵约束下的(左右)逆特征值问题,线性约束问题,或(方程组)最小二乘问题。这是解决问题的关键,是这部分区别于其他子矩阵约束问题的特色所在。在此基础上,彻底地解决了中心主子阵约束下的矩阵方程问题,得到了问题有解的充要条件,通解表达式,相应问题的最佳逼近解以及数值算例。
2.研究了线性流形上复对称矩阵,复双对称矩阵的最小二乘问题,利用复矩阵的奇异值分解,广义逆等得到了问题通解的表达式以及最佳逼近解。讨论了Hermite正定矩阵的最小二乘问题,通过对Hermite正定矩阵分块形式的分析,得到了矩阵存在最小二乘解及其最佳逼近解的充要条件,给出了最小二乘解和最佳逼近解的表达式,并提供了求最佳逼近解的算法和数值算例。
3.研究了Hermite自反矩阵的逆特征值问题、线性约束问题和Hermite反自反矩阵的逆特征值问题、线性约束问题。通过分析这两类矩阵特征向量的性质,巧妙合理地给出了逆特征值问题的数学描述。接着定义一种新的内积,利用Hermite(反)自反矩阵特有的结构,以及它们与对称向量,反对称向量的关系,获得了问题可解的充要条件,通解表达式,最佳逼近解以及数值算例。
4.讨论了行(列)对称矩阵,行(列)反对称矩阵,行(列)延拓矩阵的性质。研究了矩阵方程X~HAX=B的行(列)反对称问题,行(列)延拓矩阵的最小二乘问题,得到了问题有解的充要条件,通解表达式以及最佳逼近解。研究了求矩阵方程AXB=C的行(列)对称解或行(列)对称最小二乘解的两类迭代法,并对其中的一类迭代法进行了收敛性分析。最后证明了只需对算法稍加修改,则均可求得相应问题的最佳逼近解。
此博士论文得到了国家自然科学基金(10571047)和高等学校博士学科点专项科研基金(20060532014)的资助。
此博士论文用L~AT_EX2_ε软件打印。
The constrained matrix equation problem is to find solution to a matrix equation in a constrained matrix set.The different constrained condition,or the different matrix equation makes a different constrained matrix equation problem. More and more different constrained matrix equation problems arise out of the developments in science and engineering,especially in control theory,information theory,vibration theory,system identification,structural dynamics model updating problem,and mechanical system simulation.Thus the research results on these problems have useful applications.This dissertation considers inverse eigenvalue problem(AX=X∧),linear constraint problem and least squares problem for matrix equation AX=B,which are:
1.The research studies some real matrix equations problems under central principal submatrices constraint,including inverse eigenvalue problem,linear constraint problem,least squares problem,right and left inverse eigenvalues problem (i.e.AX=X∧,Y~TA=ΓY~T),and least squares problem of matrix equations (AX=Z,Y~TA=W~T) for centrosymmetric matrices under central principal submatrices constraint;inverse eigenvalue problem,linear constraint problem,and least squares problem for bisymmetric matrices under central principal submatrices constraint;least squares problems for skew symmetric and persymmetric matrices, symmetric and skew persymmetric matrices,and antisymmetric and skew persymmetric matrices under central principal submatrices constraint.By analysis of special properties and structure of these matrices and themselves central principal submatrices,the paper obtains that the submatrices having the same symmetric properties and structure as the given matrices,and converts these problems to (right and left) inverse eigenvalue problems,linear constraint problems,(matrix equations) least squares problems of half-sized real matrices under submatrices constraint. This simplifies and is crucial to solve the problems,and is a special feature of this part different from other papers about submatrix constraint problems.Base on these,it derives necessary and sufficient conditions for the solvability,representation of the general solutions,corresponding optimal approximation solutions and some numerical examples.
2.The research studies least squares problems for complex symmetric matrices and complex bisymmetric matrices on linear manifold,and obtains representation of general solutions and the corresponding optimal approximation solutions by using the singular value decomposition and Moore-Penrose generalized inverse. Moreover,the paper discusses least squares problem of Hermitian positive definite matrices,induces necessary and sufficient conditions and representation of least squares solutions and the optimal approximation solution by analysis of its block form,and gains a numerical method and a numerical experiment for optimal approximation solution.
3.The research studies inverse eigenvalue problems,linear constraint problems for Hermitian reflexive matrices and Hermitian skew reflexive matrices.It presents mathematical description of inverse eigenvalue problems subtly and reasonably by analysis of eigenvectors of these two kinds matrices.By using a new inner product,special structure of Hermitian(skew) reflexive matrices,and the relationship between them and reflexive vectors or anti-reflexive vectors,this paper obtains necessary and sufficient conditions for the solvability,representation of the general solutions,and corresponding optimal approximation solutions.
4.The research studies the properties of row(column) symmetric matrices, row(column) skew symmetric matrices and row(column) extended matrices,and considers linear constraint problem of matrix equations X~HAX=B for row(column) skew symmetric matrices,least squares problem for row(column) extended matrices.It derives necessary and sufficient conditions for the solvability,representation of the general solutions and corresponding optimal approximation solutions. This dissertation also puts forward two kinds of iteration methods for achieving the general solutions or least squares solutions to matrix equation AXB=C for row(column) symmetric matrices,and obtains the convergence rate of one iteration method of them.Moreover,the related optimal approximation solutions can be gained with the method which only needs to be made slight changes.
This dissertation is supported by the National Natural Science Foundation of China(10571047) and Doctorate Foundation of the Ministry of Education of China(20060532014).
This dissertation is typeset by software L~AT_EX2_ε.
引文
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