Iterative Methods For Several Constrained Matrix Equation Problems

The constrained matrix equation problem is to find solutions of a matrix equation in a set of matrices which satisfies some constraint conditions. When the constrained conditions are different, or the matrix equations are different, we can get different constrained matrix equation problems.The constrained matrix equation problems have been widely used in structural design, parameter identification, biology, electricity, molecular spectroscopy, solid mechanics, automatic control theory, vibration theory, finite elements, linear optimal control and so on.The iterative methods of the following problems which will be mainly discussed in the M.S thesis:ProblemⅠGiven A, B∈Rm×n, find X∈S, such that AX = BProblemⅡWhen ProblemⅠis consistent, let S E denote the set of its solutions, given X 0∈Rn×n, find X?∈SE, such thatWhere S is a subset of R n×n satisfying some constraint conditions.The main achievements are as follows:1. When S is the set of ortho-(anti-)symmetric matrices, the orthogonal projection iteration for the solutions of ProblemⅠby using the character and construction of this kind of matrices is given. Then the convergence of the method is proved by using the relationship between this kind of matrices and (anti-)symmetric matrices, and the estimation of the convergence rate is given. If the equation is consistent, the least-norm solution of ProblemⅠcan be obtained by the method. The iterative method of ProblemⅡcan also be obtained with the above method which only need to be made slight changes. Finally the numerical example is given, so the effectiveness of the algorithm is verified.2. When S is the set of symmetric ortho-(anti-)symmetric matrices, the orthogonal projection iteration for the solutions of ProblemⅠis given at first. Then the convergence of the method is proved by making some equivalence transformation of the matrix equation AX = B in ProblemⅠ, and the estimation of the convergence rate is given. If the equation is consistent, the least-norm solution of ProblemⅠcan be obtained by the method. The iterative method of ProblemⅡcan also be obtained with the above method which only need to be made slight changes. Finally the numerical example is given, so the effectiveness of the algorithm is verified. 3. When S is the set of anti-symmetric ortho-(anti-)symmetric matrices, the orthogonal projection iteration for the solutions of ProblemⅠis given, the convergence of the method is proved, the estimation of the convergence rate is given. If the equation is consistent, the least-norm solution of ProblemⅠcan be obtained by the method. The iterative method of ProblemⅡcan also be obtained with the above method which only need to be made slight changes. Finally the numerical example is given, so the effectiveness of the algorithm is verified.