Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X+∑i=1mAiTXδiAi=I

Abstract

In this paper necessary and sufficient conditions for the matrix equation $X+{\sum }_{i=1}^m{A}_i^T{X}^{{\delta }_i}{A}_i=I$ to have a positive definite solution $X$ are derived, where , $I$ is an $n\times n$ identity matrix, $A_i$ are $n\times n$ nonsingular real matrices, and $m$ is an odd positive integer. These conditions are used to propose some properties on the matrices $A_i$, $i=\mathrm{1,2},\dots ,m$. Moreover, relations between the solution $X$ and the matrices $A_i$ are derived.