Some properties of the nonlinear matrix equation

In this paper, the nonlinear matrix equation X s + A ⁎ X − t A = Q is investigated, where Q is a Hermitian positive definite matrix. We consider three cases of this equation: the case s , t > 0 , the case s ⩾ 1 , 0 < t ⩽ 1 and the case 0 < s ⩽ 1 , t ⩾ 1 . In the case s , t > 0 , we derive necessary conditions and sufficient condition for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. As compared to earlier works on these topics, the results we present here are more general, and the analysis here is much simpler. In the cases s ⩾ 1 , 0 < t ⩽ 1 and 0 < s ⩽ 1 , t ⩾ 1 , necessary conditions for the existence of a Hermitian positive definite solution is given, which is sharper than that of Cai and Chen (2010) [9].