Some Matrix Equations Problems Under Special Linear Constraint And Their Optimal Approximation Problems

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.