Solutions and perturbation analysis for the nonlinear matrix equation X+∑i=1mAi∗X-1Ai=I

In this paper, we study the nonlinear matrix equation X+∑i=1mAi∗X-1Ai=I arising in some physical calculations. Some new necessary and sufficient conditions for the existence of positive definite solutions are given. Based on the matrix differentiation, a precise perturbation bound for the maximal positive definite solution is derived, and the new result is illustrated by a numerical example in the end.