ON THE NONLINEAR MATRIX EQUATION $X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q$ (0＜q≤1)
Xiaoyan Yin, Ruiping Wen, Liang Fang · Bulletin of the Korean Mathematical Society · 2014
In this paper, the nonlinear matrix equation $$X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q(0 < q{\leq}1)$$ is investigated. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Two iterative methods for the maximal positive definite solution are proposed. A perturbation estimate and an explicit expression for the condition number of the maximal positive definite solution are obtained. The theoretical results are illustrated by numerical examples.