Research For Constrained Matrix Equation Based On Orthogonal Projection Iterative Methods

The constrained matrix equation problem is to find solutions of a matrix equation or a system of matrix equations in a set of matrices which satisfies some constraint conditions. Actually, it has been widely used in many fields such as structural design, biology, molecular spectroscopy, vibration theory, finite elements, and so on. The study of it has been a hot topic in the filed of numerical algebra in recent years.The master’s thesis mainly discusses the following problems:Problem I Given A∈Cm×n, B∈Cm×m, S(?)Cn×n.Find X∈S, such that AXAH=B.Problem (?) When problem I is consistent, let SE denote the set of its solutions, for given X∈Cn×n, find X∈SE, such thatWhere (?) is Frobenius norm, S is the row symmetric matrix (the row anti-symmetric matrix) or the line symmetric matrix (the line anti-symmetric matrix).When S is the row symmetric matrix (the row anti-symmetric matrix) or the line symmetric matrix (the line anti-symmetric matrix)., this dissertation discusses iterative methods of problem Ⅰ and problem Ⅱ. Based on the ideas of the orthogonal projection, the nature of the characteristics of the row (line) symmetric matrix, singular value decomposition and the orthogonal invariance of the F-norm of the matrix, firstly, orthogonal projection iterative algorithm is constructed and the convergence of the algorithm is proved. In addition, the convergence rate of the algorithm is estimated. Finally, the numerical experiments are given which indicate that the feasibility and the effectiveness of this algorithm.