Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: nonlinear matrix equation, positive definite solution, perturbation estimate, condition number

MSC numbers: 15A24, 15A45, 65H05