On the nonlinear matrix equation $X+sum_{i=1}^{m} A_{i}^{*}X^{-q}A_{i}=Q (0 < q leq 1 )$

摘要

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.