| 矩阵方程X＋A^XA=I（r〉1）正定解 |
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| 引用本文： | 姚国柱,田时宇.矩阵方程X＋A^XA=I（r〉1）正定解[J].邵阳学院学报(自然科学版),2012,9(4):1-4. |
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| 作者姓名： | 姚国柱 田时宇 |
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| 作者单位： | 长沙理工大学数学与计算科学学院,湖南长沙,410114 |
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| 基金项目： | 湖南省教育厅科学基金资助项目(10C370) |
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| 摘 要： | 本文讨论矩阵方程X＋AX^’A=I（r〉1）的（半）正定解，首先利用Brouwer不动点定理分别给出在条件AA≤I和AA〉I下该方程正定解和半正定解的存在性以及解的范围，其次利用压缩映射原理，给出方程存在唯一正定解的两个充分条件，最后得到了在A正规的情形下方程正定解的存在性．
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| 关 键 词： | 非线性矩阵方程 正定解 Brouwer不动点定理 |
The Positive Definite Solution of Matrix Equation X+A~*X~rA=I(r>1) |
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| YAO Guo-zhu,TIAN Shi-yu.The Positive Definite Solution of Matrix Equation X+A~*X~rA=I(r>1)[J].Journal of Shaoyang University:Science and Technology,2012,9(4):1-4. |
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| Authors: | YAO Guo-zhu TIAN Shi-yu |
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| Institution: | (College of Mathematics and Computing Science,Changsha University of Science & Technology,Changsha,Hunan 410114,China) |
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| Abstract: | The poisitive definite solution （pds） of matrix equation X ＋ A＊X^rA = I（r 〉 1） is discussed. First, by applying the fixed point theorem, we derive the existence of the pds solution to the matrix equation under the conditions A ＊A 〈 I and A＊A 〉 I , respectively. Second, two sufficient conditions for the equation having a unique pds are presented. Last, the existence of the equation having pds is obtained under the condition that A is normal. |
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| Keywords: | nonlinear matrix equation pds solution brouwer fixed point theorem |
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