On the nonlinear matrix equation Xp = A#t (MT (X –1 + B)–1 M)

Abstract

ABSTRACT In this paper, the nonlinear matrix equation , where is a positive integer, , M is an nonsingular matrix, A is a positive definite matrix and B is a positive semidefinite matrix, is considered. The notation is the t-weighted geometric mean of the positive definite matrices A and B. Based on the properties of the Thompson metric, we prove that the nonlinear matrix equation always has a unique positive definite solution and we compare it with the unique positive definite solution of the equation , which has been studied in Jung et al. [On the solution of the nonlinear matrix equation . Linear Algebra Appl. 2009;430:2042–2052]; Meng and Kim [The positive definite solution of the nonlinear matrix equation . J Comput Appl Math. 2017;322:139–147]. A fixed-point iteration method for obtaining the unique positive definite solution and an elegant estimate of the solution are given. Perturbation analysis of the unique positive definite solution is presented.