Property of an orthogonal projection matrix
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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.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.
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Key words:
- orthogonal projection matrix /
- singular value /
- unitary matrix /
- compound matrix
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[1] {1}张杰, 杨春德. 正交投影矩阵的一个求法[J]. 重庆邮电学院学报:自然科学版, 2006(01):141-142.{2}庞善起. 一类正交投影矩阵及其相关正交表[J]. 应用数学学报,2005(04):668-674.{3}李乔. 矩阵论八讲 [M]. 上海: 上海科学技术出版社, 1988.{4}詹兴致. 矩阵论[M]. 北京: 高等教育出版社, 2008.{5}ZHAN X Z. Matrix Inequalities[M]. Berlin: Spring-Verlag, 2002.{6}STEWART G W, SUN J G. Matrix Perturbation Theory[M]. [S.L.]:Academic, 1990.
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