ON THE NONLINEAR MATRIX EQUATION X+∑ᵢ₌₁ᵐAᵢ^*X-qAᵢ=Q (0＜q≤1)

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In this paper, the nonlinear matrix equation X+∑ᵢ₌₁ᵐAᵢ^*X-qAᵢ=Q(0 < q≤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.