D Symmetric(Antisymmetric) Solutions Of Matrix Equation AXB=C And Its Optimal Approximation

The constrained matrix equation problem is to find solutions or least-squares solutions and its corresponding optional approximation of a matrix equation under some restricted conditions . It has been widely used in structural design, parameter identification, nonlinear programming, finite elements, biology, solid mechanics and automatic control theory and so on.The master's degree thesis considers D symmetric and D antisymmetric solutions of matrix equation AXB = C, which are described as follows:Problem 1 : Given matrices A∈R^m×n, B∈R^n×m, C∈R^m×m, find a D symmetric (antisymmetric) matrix X∈R^n×n such that AXB = C.Problem 2 : Given matrices A∈R^m×n, B∈R^n×m, C∈R^m×m, find a D symmetric (antisymmetric) matrix X∈R^n×n such that ||AXB - C||_F = min.Problem 3 : Let E₂ denote the solution set of Problem 2, given(?)∈R^n×n, find (?)∈E₂ such thatThe main results of the thesis are as below:1. Applying the generalized singular value decomposition for matrix pair, we obtain solvable conditions and general solution expression for problem 1. We also get the expression and the special structure of least-squares D symmetric solutions.2. Applying generalized conjugate gradient method, we design a iteration method for computing D symmetric solutions of Problem 1 and 2. We prove that the iterative method is convergence within finite steps. By choosing the initial matrix, the method can compute the minimal norm solution. Furthermore, we successfully transform the optimal approximation problem into the minimal norm problem and obtain the method to compute the unique optimal approximation solution. The numerical example shows that the algorithm is efficient.3. Applying the canonical correlation decomposition for matrix pair, we obtain the necessary and sufficient conditions for AXB = C having a D antisymmetric solution X and the general solution expression. By projection, we transform successfully least-squares problem into the consistent problem and get the general least-squares solutions. 4. Applying generalized conjugate gradient method, we design an algorithm to compute the D antisymmetric solutions for the consistent matrix equations and least-squares D antisymmetric solutions for the inconsistent matrix equation, the algorithm also can compute the optimal approximation solution. The numerical example shows that the algorithm is efficient.This thesis is supported by the National Natural Science Foundation(10571047) of China.