The Hermitian Solutions Of The Nonlinear Matrix Equation

In the modern science and engineering calculations, we could often meet the problem of solving the nonlinear matrix equation. In the modern industry, especially in the application of modern physics, solving and analysis this type of equation is increasingly important. To solving the nonlinear matrix equation has gradually become an important topic in the field of modern computational mathematics.In this paper, we consider the nonlinear matrix equation Y-N*Y-1N=Q. where Q is the n*n Hermitian symmetric positive definite matrices, and N*is the conjugate transpose of the matrix N. In the introduction, we not only give out the background and the history of this type of nonlinear matrix equation, but also show the current research situation and purpose. The first chapter is the preface. In this part, we give out the symbols and marks as well as related prior knowledge and Theorem theory.Chapter Ⅱ and Ⅲ are the main parts of this paper. The second chapter mainly talks about the Hermitian positive definite solution of the equation. Section2.1gives two necessary and sufficient conditions while the equation has the unique Hennitian positive definite solution. Section2.2gives the proof of the relations forms and Section2.3proves that the equation has the unique Hermitian positive definite solution. Chapter Ⅲ mainly talks about the Hermitian negative definite solution of the equation. Section3.1gives the proof of the existence of the equation’s negative definite solution. Section3.2gives two necessary and sufficient conditions for the Hermitian negative definite solution of the equation. Section3.3discusses the relationship between the Hermitian negative definite solution and the coefficient matrix of the equation. Section3.4proves that the equation has the unique Hermitian negative definite solution.The fourth chapter is the numerical experiment. It proves that the iterative algorithm convergence to this equation’s solution.