The Iterative Method For The Solutions And The Optimal Approximation Of The Constrained Matrix Equation

The constrained matrix equation problem is to find solutions of a matrix equation or a system of matrix equations in a set of matrices systisfing some constraint conditions. When the constraint conditions are different, or the matrix equation is different, we can get a different constrained matrix equation problem.The constrained matrix equation problem have been widely used in structural design, parametre identification, biology, electricity, molecular spectroscopy, solid mechanics, automatic control theory, vibration theory, finite elements, linear optimal control and so on.This Ph.D. thesis considers the following problems systematically.Problem I Given , find , such thatAX = B.Problem II Given ,, find such thatAXB = C.Problem III Given find such thatProblem IV When Problem I or II or III is consistent, let Se denote the set of its solutions, for given , find , such thatwhere is Frobenius norm, S is Rn×p or a subset of Rn×p (n = p) satisfying some constraint conditions, such as symmetric, skew-symmetric, centrosymmet-ric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric.The main achievements are as follows:1. For problem I, many references have obtained a series important result by means of matrices decompositions. In this thesis, we consider this problem in different way with the help of the method of convergence of conjugate. For the first time, we use iterative methods successfully in finding its symmetric solutions, skew-symmetric solutions, centrosymmetric solutions, centroskew symmetric solutions,reflexive matrix solutions, antireflexive matrix solutions, bisymmetric solutions, symmetric and antipersymmetric solutions and its optimal approximation constrained solution. This is an exciting development of matrix theories and matrix research methods.2. For problem II, many references have studied it and obtained its common solutions, symmetric solutions, skew-symmetric solutions and its optimal approximation constrained solution, but the representation of its solutions are complicated. Since now, the problems to find its centrosymmetric solutions, centroskew symmetric solutions, reflexive matrix solutions, antireflexive matrix solutions and its optimal approximation constrained solution haven't been solved by means of matrices decompositions, and it is more difficult to find its two-constrained solutions and its optimal approximation two-constrained solution, such as bisymmetric, symmetric and antipersymmetric. For the first time, we apply iterative methods to study how to find its constrained solutions, as the result of this, we have obtained its centrosymmetric solutions, centroskew symmetric solutions, reflexive matrix solutions, antireflexive matrix solutions, bisymmetric solutions, symmetric and antipersymmetric solutions and its optimal approximation constrained solution with flying colours, this is the useful improvment for the achievements of references.3. For problem III, many references have solved the problems of finding constraint solutions for some special kinds of matrix equations, and there are some difficulties to find its optimal approximation constrained solution. So far, the effective methods have not been discovered to find the constraint solutions for matrix equations A1X B1 =C1, A2XB2 = C2, such as symmetric solutions, skew-symmetric solutions, centrosymmetric solutions, centroskew symmetric solutions, reflexive matrix solutions, antireflexive matrix solutions, bisymmetric solutions, symmetric and antipersymmetric solutions and its optimal approximation constrained solution. For the first time, we study these problems by use of iterative methods, and solve them successfully, this is the important improvement for the achievements of references.The disadvantage of the iterative methods constructed in this thesis is that we can ignore the verification to the consistent conditions for the matrix equation or matrix equations, instead of this, the solvability of the problem I or II or III can be determined automatically during the iteration...