Abstract: The matrix equation X + A ^* X^-2 A = I is studied. Some necessary and sufficient conditions for the existence of a positive definite solution of this equation are derived. Based on them, some properties of the coefficient matrix A are presented when the matrix equation has a positive definite solution.

Key words: nonlinear matrix equation, positive definite solution, necessary and sufficient condition

摘要： 给出了矩阵方程X + A ^* X^-2 A = I有对称正定解的两个充分必要条件，它们在算法设计和理论分析上可能有一定的用途。根据这两个定理，当矩阵方程有对称正定解时，给出了系数矩阵A必须满足的条件，这些条件大部分都是很容易验证的。

关键词: 非线性矩阵方程, 对称正定解, 充分必要条件