Some investigation on Hermitian positive definite solutions of the matrix equation Xs+A∗X-tA=Q

AbstractIn this paper, the Hermitian positive definite solutions of the matrix equation Xs+A∗X-tA=Q are considered, where Q is an Hermitian positive definite matrix, s and t are positive integers. Necessary and sufficient conditions for the existence of an Hermitian positive definite solution are derived. A sufficient condition for the equation to have only two different Hermitian positive definite solutions and the formulas for these solutions are obtained. In particular, the equation with the case AQ12=Q12A is discussed. A necessary condition for the existence of an Hermitian positive definite solution and some new properties of the Hermitian positive definite solutions are given, which generalize the existing related results