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

TL;DR

This paper investigates conditions for the existence of positive definite solutions to a complex nonlinear matrix equation, establishing equivalences, bounds, and criteria that advance understanding of such equations in matrix analysis.

Contribution

It introduces a matrix operator approach to relate complex and real matrix equations, providing new existence criteria, bounds, and conditions for positive definite solutions.

Findings

Existence of solutions is equivalent to a real matrix equation.

Maximal and minimal solutions exist if solutions are present.

Bounds for solutions are established based on matrix A.

Abstract

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.