  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.
