华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (1): 97-99.

• 应用数学与基础数学 • 上一篇    下一篇

正交投影矩阵的一个性质

杜 琨1, 顾桂定2   

  1. 1.上海财经大学 金融学院, 上海 200433; 2.上海财经大学 数学系, 上海 200433
  • 收稿日期:2011-04-01 修回日期:2011-07-01 出版日期:2012-01-25 发布日期:2012-01-26

Property of an orthogonal projection matrix

DU Kun 1, GU Gui-ding 2   

  1. 1. School of finance, Shanghai University of Finance and Economics, Shanghai} 200433,  China; 2. Department of Mathematics, Shanghai University of Finance and Economics, Shanghai} 200433,  China.
  • Received:2011-04-01 Revised:2011-07-01 Online:2012-01-25 Published:2012-01-26

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摘要： 证明了秩为~$k$~的正交投影矩阵, 一定存在~$k$~阶主子阵, 其~Rayleigh~商有一个正的下界. 证明中综合使用了矩阵的奇异值、特征值、范数之间的优超关系以及酉矩阵和复合矩阵的性质, 为进一步揭示正交投影矩阵的性质提供了一种可能.

关键词: 正交投影矩阵, 奇异值, 酉矩阵, 复合矩阵

Abstract: In this paper we showed that for an orthogonal projection matrix with rank $k$, there exists an principal submatrix with order $k$ of the matrix, such that its Rayleigh quotient has a positive lower bound. The proof was made by using the relation of the singular values, eigenvalues and norm of matrices, as well as the properties of unitary matrix and compound matrix.

Key words: orthogonal projection matrix, singular value, unitary matrix, compound matrix

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