On the Hermitian positive definite solution of the nonlinear matrix equation X+∑i=1mAi∗X-1Ai=I

In this paper, we study the matrix equation X+∑i=1mAi∗X-1Ai=I, where Ai (i = 1, 2, …, m) is square matrices, and obtain some conditions for the existence of positive definite solution of this equation. Two iterative algorithms to find the maximal positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms.