A Kind Of Necessary And Sufficient Conditions For The Existence Of A Positive Definite Solution Of The Function X^s + A^* X^-tA =Ⅰand Its Estimation

Matrix equation is a very important branch in the matrix theory. In partic-ular, the nonlinear matrix equation has important applications in control theory,ladder networks and dynamic programming. Therefore, many scholars pay muchattention to studying nonlinear matrix equation.In this paper, we mainly discuss several necessary and su(?)cient existenceconditions of a Hermitian positive definite solution for a class of nonlinear matrixequations. If the positive solution of this equation exists, we estimate the positivesolution, the maximum and minimum eigenvalues of the equation, which is betterthan some existing results. This paper is divided into four parts:In the first part, we introduce the applied background, the recent results anditerative algorithms of this nonlinear matrix equation. In addition, we give somenotations and definitions.In the second part, applying the properties of positive definite matrix andnormal matrix, utilizing contragradient transformation and unitary decomposi-tion, we present some necessary and su(?)cient existence conditions of a positivedefinite solution for this nonlinear matrix equation. Further, we give a numericalexample to show its e(?)ectiveness.In the third part, applying the results derived in the second part, usingunitary invariant of spectral norm and the classical eigenvalue inequality, we showthe range of the maximum and minimum eigenvalues for this nonlinear matrixequation. Finally, the numerical example illustrates its superiority.In the fourth part, based on the previous results and the eigenvalue boundsdeduced in the second part, we propose new upper and lower bounds of thepositive definite solution for this nonlinear matrix equation. At last, we provethat our bounds are tighter than some of the existing results in theory.