# ON THE NONLINEAR MATRIX EQUATION $X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q$(0＜q≤1)

• Yin, Xiaoyan ;
• Wen, Ruiping ;
• Fang, Liang
• Received : 2013.03.27
• Published : 2014.05.31
• 64 7

#### Abstract

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

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