Research On The Iteration Method For Several Kinds Of Consistent And Inconsistent Constrained Matrix Equation

The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. The study of it has been a hot topic in the field of numerical algebra in recent years. Actually, it is widely used in many fields such as structural design, system identification, automatics control theory, vibration theory.This dissertation considers mainly the following problems.Problemâ… Given A,B∈R^m×n,S(?)R^n×n, find X∈S, such thatAX=B or ||AX-B||=minProblemâ…¡GivenA∈R^m×n,B∈R^n×p,C∈R^m×p,S(?)R^n×n, find X∈S, such thatAXB=C or ||AXB-C||=minProblemâ…¢Given A₁∈R^m₁×n,B₁∈R^n×p,C₁∈R^m₁×p,A₂∈R^m₂×n,B₂∈R^n×p,C₂∈R^m₂×p,S(?)R^n×n, find X∈S, such thatProblemâ…£Let S_E denote the solution set of above problems, for given X₀∈S(?)R^n×n, find (?)∈S_E, such thatProblemâ…¤Given, find [X₁,X₂,…,X_l](where X_i∈S_i,i=1,2,…,l), such thatA₁X₁B₁+A₂X₂B₂+…+A_lX_lB_l=CProblemâ…¥When Problemâ…¤is consistent, let SE denote the set of its solutions, for given (where),find, such that where||·|| is Frobenius norm, S and S_i are the matrix set satisfying some constraint conditions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric. The main works and creative point on this dissertation are as follows.1. The concept of gradient matrix in subspace is presented for the first time.2. The computational formula of the gradient matrix in subspace is obtained by applying Taylor series expansions, decomposition theorem of space and projective theorem (The gradient matrixs in different subspace are different).3. The definition of generalized conjugate (or M-conjugate) is introduced and the generalized conjugate gradient method is presented.4. The solutions of Problemâ… ,â…¡andâ…¢are discussed by using the generalized conjugate gradient method. When the equation (equations) is consistent, the solutions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric are successfully found; When the equation (equations) is inconsistent, the least-squares solutions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric are also found successfully.The generalized conjugate gradient method has the following traits:(1) It can judge automatically the information of solutions. When the equation (equations) is consistent, it converges the solutions of the equation (equations). When the equation (equations) is inconsistent, it converges the least-squares solutions of the equation (equations).(2) For any initial matrix, a solution can be obtained within finite iteration steps in the absence of roundoff errors. If a special kind of initial matrix is chosen, the solution with least norm can be obtained, and wonderfully handle Problemâ…£5. An iterative method is constructed to find the solution group of Problemâ…¤. The solution group such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric are successfully found.This dissertation is supported by the Natural Science Foundation of China(105 71047).This dissertation is typeset by software L^AT_EX2_ε.