Positive Definite Solutions of the Nonlinear Matrix Equation X+AHXˉ−1A=IX+A^{\mathrm{H}}\bar{X}^{-1}A=I

This paper is concerned with the positive definite solutions to the matrix equation $X+A^{\mathrm{H}}\bar{X}^{-1}A=I$ where $X$ is the unknown and $A$ is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation $W+B^{\mathrm{T}}W^{-1}B=I$ which has been extensively studied in the literature, where $B$ is a real matrix and is uniquely determined by $A.$ It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix $A$. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation X

Nonlinear matrix equation Xs+A∗X−t1A+B∗X−t2B=Q has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation Xs+A∗X−t1A+B∗X−t2B=Q are considered, where Q is a Hermitian positive definite matrix, A, B are nonsingular complex matrices, s is a positive number, and 0<ti≤1, i=1,2. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method