Thompson metric method for solving a class of nonlinear matrix equation

Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation Xq-A∗F(X)A=Q(q>1)Xq-A∗F(X)A=Q(q>1) always has a unique positive definite solution. An iterative method is proposed to compute the unique positive definite solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique positive definite solution is derived in the end.