\documentstyle[11pt]{article} \textwidth 6.6in \textheight 8.0in \oddsidemargin 0in \evensidemargin 0in \topmargin 0.1in \parindent 0.5in \begin{document} \thispagestyle{empty} \centerline {\Large\bf Positive Definite Solutions } \centerline {\Large\bf of the Equation $X - A^* X^{-2} A = I$} \vspace{0.4cm} \centerline {\Large\bf Ivan G. Ivanov, Borislav V. Minchev, Vezhdi I. Hasanov } \vspace{0.4cm} \rm In this paper we study the positive solutions of the matrix equation $X - A^* X^{-2} A = I$. Necessary and sufficient conditions for existence of positive definite solutions of this equation are derived. Algorithms for numerical computation of such solutions are given. Sufficient conditions for convergence of the iterative methods \begin{eqnarray} X_0 = \alpha I, \ \ X_{k+1} &=& \sqrt {A (X_k - I)^{-1} A^*}, \ \ \ k=0,1,2, \ldots \ \ \ ( \alpha > 2 ) \\ X_0 = I, \ \ X_{k+1} &=& I + A^* X_k^{-2} A, \ \ \ k=0,1,2, \ldots \end{eqnarray} are presented. If there exsists a real number $\alpha$ for wich the following conditions are satisfied $$A A^* < \alpha^2(\alpha-1) I, \ \ \ \sqrt {\frac{A A^*}{\alpha-1}} - \frac{1}{\alpha^2} A^* A > I$$ or $$A A^* > \alpha^2(\alpha-1) I, \ \ \ \sqrt {\frac{A A^*}{\alpha-1}} - \frac{1}{\alpha^2} A^* A < I$$ then the matrix sequence obtained by formula $(1)$ is convergent to a positive definite solution of the equation. If the matrix $A$ is normal $( A A^* = A^* A )$ and the spectral norm satisfies $$\|A\| < \sqrt {\frac{\sqrt{3}-1}{2}},$$ then the matrix sequence obtained by formula $(2)$ is convergent to a positive definite solution of the equation. Numerical experiments for computing of positive definite solutions of the considered equation are described. The results from the experiments show the effectiveness of the iterative methods for the described examples. \end{document}