Perturbation analysis of the Hermitian positive definite solution of the matrix equation X − A*X−2A = I

Consider the nonlinear matrix equation X − A*X−2A = I, where A is an n × n complex matrix, I the identity matrix and A* the conjugate transpose of a matrix A. In this paper, it is proved that this matrix equation has a unique Hermitian positive definite solution provided ∥A∥2 < 1, and moreover, under the condition ∥A∥2 < 1, a perturbation bound for the Hermitian positive definite solution to this matrix equation is derived, and an explicit expression of the condition number for the Hermitian positive definite solution is obtained. The results are illustrated by using some numerical examples.