The Least-squares Problem Of Centrosymmetric Matrices And Its Approximation

The constrained matrix equation problem is a problem 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, sys-tem identification, structural dynamics model updating problem, and mechanical system simulation. Thus the results of these problems have useful applications.This thesis considers the linear constraint problem and least squares problem for matrix equation(AX=B)and matrix equations(AX=Z, YTA=WT), which are:1. The research studies the constrained matrix equation problem of a kind of cen-trosymmetric matrices,including linear constraint problem, least squares problem and optimal approximation problem. By analysis of the special properties of a kind of cen-trosymmetric matrices, the paper derives necessary and sufficient conditions solutions for the solvability, representation of the general solutions, corresponding optimal approxi-mation solutions and gives some numerical examples.2. The research studies matrix equation problem for matrix equation(AX=B)under central principal submatrices constraint with orders×t. By analysis of the special proper-ties and structure of a kind of centrosymmetric matrices and themselves central principal submatrices constraint with orders×t, the paper obtains that the submatricess has the same symmetric properties and structure as the given matrices.Base on these, it derives necessary and sufficient conditions for the solvability, representation of the general so-lutions,corresponding optimal approximation solutions and gives some numerical exam-ples.3. The research studies least squares problems of the matric equations(AX=Z, YTA= WT)under central principal submatrices constraint with orders×t. The least-squares problems are transformed into equation problems at first, then by using the singular value decomposition and generalized singular value decomposition method, the general repre-sentation of the least-squares solutions, the optimal approximation solutions, numerical algorithms and examples are given.