The Theory And Algorithm Research For Solving Several Classes Of Constrained Matrix Equation

The constrained matrix equations have been widely applied in system engineering, autocontrol theory, economics, network programming, civil struction engineering and vibratory theory. The problems we will mainly discuss in the Ph.D. Thesis are as follows:Problem I Given two matrices A,B ∈ Rm×n, a constrained set S Rn×n. Find X ∈ S such thatAX - B = min.Problem II Given two matrices X, B 6 Rm×n, a constrained set S Rm×m. Find A e S such thatAX=B.Problem III Given an eigenvalue matrix ∈ Rk×k which is diagonal or 2 x 2 block diagonal, and the corresponding eigenvector matrix X ∈ Rn×k a constrained set S Rn×n. Find A ∈ S such thatAX = XAorProblem IV Given blocks , and a constrained set S Rn×n. Find a block A22 ∈ Gm2×m2 such thatProblem IV Given A* ∈ Rn×n, denote SE by the solution set of the above problems respectively. Find a solution matrix A ∈ SE such thatThe main fruits of the paper are as below: The numerical solutions for Problem I on the closed convex cone are sys-temically studied in the paper. Applying the approximation theory of closed convex cone and convex analysis in the creative way, the properties of least-squares solution are obtained. Using optimal theory, a gradient projection iteration is proposed to compute approximately the solutions of Problem I ona closed convex cone. The global convergence and linear convergence have been theoritically proved. For eight familiar closed convex cones, a series of MATLAB procedures are provide to compute the solutions for them easily and conveniently.2. We discuss firstly the inverse problems (i.e. Problem II) of generalized reflection matrices, skew-symmetric orthogonal matrices, partial isometries and orthogonal projectors respectively, have obtained the necessary and sufficient conditions for solution existence. Furthermore, we also consider the least-sqaures solutions for them, provide algorithms or MATLAB functions to compute the solutions on PC.3. As for Problem III, we research the inverse eigenvalue problem of Hamiltonian matrices, get the methods to compute the least-squares solution and the best approximate solution for it, provide MATLAB procedures to calculate the least-squares solution and the least-squares solution with the least norm, we also investigate the inverse eigenvalue problem of orthogonal matrices, present algorithms to compute one solution when there exists at least a solution, to compute the least-squares solutions and optimal approximate solution. Some numerical examples are provided to illustrate the theory and the algorithms.4. Problem IV is matrix completion problem. We look into the matrix completion problem of nonsingular matrices, skew-symmetric invertible matrices and orthogonal projectors, deduce firstly the necessary and sufficient conditions for solution existence, provide some MATLAB functions or algorithms to compute the solutions, least-squares solutions and optimal approximate solutions.5. We also study the approximation problem(i.e. Problem V) on the subspace, the closed bounded set and linear manifold, give algorithms and corresponding numerical examples.This Ph.D. Thesis is supported by the National Natural Science Foundation of China.This Ph.D. Thesis is typeset by LATEX2ε...