Some Properties On Positive Definite Solution Of Nonlinear Matrix Equation

Matrix equations play an important role in the field of mathematics, and its role in the development of many industries to be reckoned with. Among them, the problem of solving the matrix equation is the most important one.The matrix theory is an important branch of mathematics. It is widely used in the field of signal processing, engineering calculation, power system, stability theory, combination and graph theory and so on. With the development and continuous improvement of computer and network, the matrix theory has a larger play space, and it also has a good prospect. Which will causes the great attention of the scholars in various fields who achieved many good results by studying.In this thesis, I mainly discuss and prove the problems of the positive definite solution of the nonlinear matrix equationX+A*X-qA-Q and the nonlinear matrix equationXs+A*X-tA= Q, and have new conclusions.In the third chapter, the existence and uniqueness of positive definite solution of nonlinear matrix equations X+A*X-qA= Qare given, whenq≥ 1 and nonlinear matrix equationsX+A’X-qA-Q, when 0< q< 1, And I also prove the necessary condition and sufficient condition for the existence of the positive definite solution, hereA is nonsingular n order plural matrix, and Q is n order Hermite positive definite matrix.In the fourth chapter, we analysis of the positive definite solutions of existence and uniqueness of the nonlinear matrix equationsX+A*X-qA= Q, besides, the detailed proven is given by me, at the same time, the existence of the solution of this equation are given.In the third chapter, I will study the special form X+A*X-2A= I of matrix equation Xs+AX*A=Q, in this section, I discuss and prove the existence of the positive definite solution of the equation X+A*X-2A= I first. In the latter part of this chapter, I present and prove the other properties of the positive definite solution by using the positive definite solution of the equation X+A*X-2A=I.