Positive definite solutions of the matrix equations X A X􀀀qA D Q .q 1/

In this paper we investigate nonlinear matrix equations X A X􀀀qA D 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 C A X􀀀qA D Q to have two different positive definite solutions and several sufficient conditions for the equation X􀀀A X􀀀qA D 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.