Thompson metric method for solving a class of nonlinear matrix equation

Abstract

Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation X q - 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.