Some Properties On Positive Definite Solution Of Nonlinear Matrix Equation

Matrix theory is not only the basis of the classical mathematics learning, but also the mathematical theory of practical.Matrix theory is an important branch of mathematical application and It has been applied in engineering calculation, stability theory, signal processing theory, combined with the graph theory, power systems, and other fields. Especially the wide application of computer opened up a broad prospect for the application of matrix theory. It caused scholars at home and abroad and the attention of the whole engineering and technical personnel, and obtained a series of scientific research achievements. Nonlinear matrix equation as an important part of the matrix theory has become an active and broad field of study at present. In This article the nonlinear properties of positive definite matrix equations are studied, and on the basis of the existing research results,some new conclusions are obtained.The first part respectively summarizes some properties including nonlinear matrix equation X+A*X-qA=Q, q∈[1,∞) and X+A*X-qA=Q, q∈(0,1]。sufficient conditions of existence of the positive definite and under different conditions study existence and uniqueness positive definite solution new sufficient conditions are obtained.The second part sums up about the scope of positive definite solution of nonlinear matrix equation Xs+A*X-tA=Q, and under different conditions we optimized the original scope of the positive definite solution.