On Ando–Li–Mathias geometric mean equations

In this paper we consider a family of nonlinear matrix equations based on the higher-order geometric means of positive definite matrices that proposed by Ando–Li–Mathias. We prove that the geometric mean equationX=B+G(A1,A2,…,Am,X,X,…,X︸n)has a unique positive definite solution depending continuously on the parameters of positive definite Ai and positive semidefinite B. It is shown that the unique positive definite solutions Gn(A1, A2, … , Am) for B = 0 satisfy the minimum properties of geometric means, yielding a sequence of higher-order geometric means of positive definite matrices.