On Hermitian positive definite solutions of nonlinear matrix equation

Abstract

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equations X + A*X^−qA = Q and X − A*X^−qA = Q are discussed where q∈(0,1]. Some sufficient conditions for the existence of Hermitian positive definite solutions for these equations are derived. A sufficient condition for X + A*X^−qA = Q is given to have a unique Hermitian positive definite solution. The perturbation analysis for X − A*X^−qA = Q is discussed at last.