Research On The Special Solutions Of Matrix Equations And Their Optimal Approximation Problems

The constrained matrix equation problem is to find the solutions of a matrix equation in a constrained matrix set. In recent years, as it has been widely used in many fields such as in linear optimal control, the finite element,the principal component analysis, structure design,vibration theory, nonlinear programming,automatic control theory, the study of the constrained matrix equation problem has become one important topic in the field of computational mathematics.This paper mainly studied the orthogonal solution of matrix equation AX ?B under certain constraints of the(P,Q)- symmetric orthogonal matrix and the(P,Q)-anti-symmetric orthogonal matrix.The necessary and sufficient conditions of the matrix equation AX ?B,with(P,Q)-symmetric or(P,Q)-anti-symmetric orthogonal matrix constraints are given, and the general expressions of the solutions as well as the optimal approximation solutions are obtained. Furthermore, in order to verify the correctness of the conclusions, some algorithms and numerical experiments are reported. Finally, The solutions of the matrix equation AXB ?C,with(P,Q)-symmetric or(P,Q)-anti-symmetric orthogonal matrix constraints are given, and the general expressions of the solutions are obtained.The paper mainly concluded as follows:1. Putting forward the concept of(P,Q)- symmetric orthogonal matrix, its structure is constructed, the necessary and sufficient conditions and the general expressions of the solutions as well as the optimal approximation solutions are obtained by means of the singular value decomposition and polar decomposition.2. Putting forward the concept of(P,Q)- anti-symmetric orthogonal matrix, its structure is constructed, the necessary and sufficient conditions and the general expressions of the solutions as well as the optimal approximation solutions are obtained by means of the singular value decomposition and polar decomposition.3. The special(P,Q)-symmetric orthogonal solutions of the matrix equation AXB ?C are studied. Through constraints on the structure of the solutions, the conditions and the general expressions of the orthogonal solutions of the matrix equation AXB ?C are obtained. The problem of the-symmetric orthogonal solutions of solving the equation AXB ?C can be transformed into another equation equivalently,and by investigating the solvability of this new equation, then the solutions for the solvability of the matrix equation AXB ?C, with(P,Q)-symmetric orthogonal matrix constraints are given, and the general expressions of the solutions are obtained.4. On the basis of the conclusion three,the solutions for the solvability of the matrix equation AXB ?C, with(P,Q)-anti-symmetric orthogonal matrix constraints are given, and the general expressions of the solutions are obtained.