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

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