Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I

Abstract

We investigate the nonlinear matrix equation $X^r-{\sum }_{i=\mathrm{1}}^m{A}_i^{\mathrm{\ast }}{X}^{-{\delta }_i}{A}_i=I$, where $r$ is a positive integer and ${\delta }_i\in (\mathrm{0,1}],\hspace{0.17em}\text{for}i=\mathrm{1,2},\dots ,m$. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.