矩阵方程X + A * X-2 A = I有对称正定解的充分必要条件

北京大学学报（自然科学版）

矩阵方程X + A^X^-2 A = I*有对称正定解的充分必要条件

程明松

北京大学数学科学学院，北京，100871

收稿日期:2003-12-17 出版日期:2005-01-20 发布日期:2005-01-20

Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A^X^-2 A = I*

CHENG Mingsong

School of Mathematical Sciences, Peking University, Beijing, 100871, E-mail: mscheng@dlut.edu.cn

Received:2003-12-17 Online:2005-01-20 Published:2005-01-20

摘要/Abstract

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

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

中图分类号:

程明松. 矩阵方程X + A^*X^-2 A = I有对称正定解的充分必要条件[J]. 北京大学学报（自然科学版）.

CHENG Mingsong. Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A^*X^-2 A = I[J]. Acta Scientiarum Naturalium Universitatis Pekinensis.

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