Abstract
In this paper we investigate nonlinear matrix equations X ± A ∗ X − q A = Q where q ≥ 1 . We derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations. We provide a sufficient condition for the equation X + A ∗ X − q A = Q to have two different positive definite solutions and several sufficient conditions for the equation X − A ∗ X − q A = Q to have a unique positive definite solution. We also propose iterative methods for obtaining positive definite solutions of these equations. Numerical examples are given to illustrate the effectiveness of the methods.
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