幾類線性矩陣方程的加權(quán)最小二乘解及其最佳逼近問題.pdf

1、湖南大學碩士學位論文幾類線性矩陣方程的加權(quán)最小二乘解及其最佳逼近問題姓名：周立平申請學位級別：碩士專業(yè)：應(yīng)用數(shù)學指導教師：李董輝;鄧遠北20070427碩士學位論文 IIIAbstract The constrained linear matrix equations and related least squares problems have wide range of practical applications, includi

2、ng structure design, parameter identification, biology, automatic control theory, vibration theory, finite elements signal processing and so on. Due to this reason, the study in constrained linear matrix equation has tak

3、en good progress, and has become a welcome research topic in computational mathematics. So far, almost all existing research in matrix equation problem focuses on the case where Frobenius norm is used. In this thesis, w

4、e define a weighted Frobenius norm F W WA A = . By the use of singular value decomposition and the dual theory in Hilbert space, we study the solution of the following four problems: Problem I: Given n m R A × ∈ ,

5、 n m R B × ∈ , m m SR W × + ∈ . Find X such that min = ? w B AX , where m m SR × +denotes m-order real symmetric and positive definite matrix. Problem II: Given m n R A × ∈ , m m R B × ∈ ,

6、 m m SR W × + ∈ . Find X such that min = ? wT B XA A . Problem III: Given n n R A × ∈ * . Find E S A∈ ?such that * * min ? A A A AE S A ? = ? ∈ , where E S denotes the solution set of Problem I or Problem

7、II. The main results of this thesis are listed as follows: 1. We derive the expressions of the solution of Problem I and related optimal approximation problem. 2. We also derive the expressions of the solution of Problem

8、 II and related optimal approximation problem. 3. We discuss the least squares symmetric and anti-symmetric solutions of matrix equation D XB BT =on some linear manifold. We also study the related optimal approximation