On the Reduced Minimum Modulus of Projections and the Angle between Two Subspaces
Received:December 20, 2008  Revised:June 30, 2009
Key Word: subspace   angle between two subspaces   reduced minimum modulus.  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10871224) and the Fundamental Research Funds for the Central Universities (Grant No.GK 200902049).
Author NameAffiliation
Xiu Hong SUN School of Science, Xi'an University of Science and Technology, Shaanxi 710054, P. R. China 
Yuan LI College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China 
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      Let $\mathcal{M}$ and ${\mathcal{N}}$ be nonzero subspaces of a Hilbert space $\mathcal{H},$ and $P_{\mathcal{M}}$ and $P_{\mathcal{N}}$ denote the orthogonal projections on $\mathcal{M}$ and ${\mathcal{N}},$ respectively. In this note, an exact representation of the angle and the minimum gap of $\mathcal{M}$ and ${\mathcal{N}}$ is obtained. In addition, we study relations between the angle, the minimum gap of two subspaces $\mathcal{M}$ and ${\mathcal{N}},$ and the reduced minimum modulus of $(I-P_{\mathcal{N}})P_{\mathcal{M}}$.
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