Iterative positive definite solutions of the two nonlinear matrix equations \(X \pm A^{T}X^{-2} A =I \) (Q1774852)

Consider the matrix equation \(X + \varepsilon A^\top X^{-2}A= I\) in \(\mathbb{R}^{n\times n}\), where \(\varepsilon= \pm1\). The iteration scheme \(X_{k+1}= I -\varepsilon A^\top X^{-2}_k A\), \(X_0= \alpha I\), \(\alpha> 0\), for finding the positive definite solution is justified under certain restrictions on the coefficient matrix \(A\).