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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stiefel manifold ： ウィキペディア英語版 Stiefel manifold In mathematics, the Stiefel manifold ''V''''k''(R''n'') is the set of all orthonormal ''k''-frames in R''n''. That is, it is the set of ordered ''k''-tuples of orthonormal vectors in R''n''. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold ''V''''k''(C''n'') of orthonormal ''k''-frames in C''n'' and the quaternionic Stiefel manifold ''V''''k''(H''n'') of orthonormal ''k''-frames in H''n''. More generally, the construction applies to any real, complex, or quaternionic inner product space. In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent ''k''-frames in R''n'', C''n'', or H''n''; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group. ==Topology== Let F stand for R, C, or H. The Stiefel manifold ''V''''k''(F''n'') can be thought of as a set of ''n'' × ''k'' matrices by writing a ''k''-frame as a matrix of ''k'' column vectors in F''n''. The orthonormality condition is expressed by ''A'' *''A'' = 1 where ''A'' * denotes the conjugate transpose of ''A'' and 1 denotes the ''k'' × ''k'' identity matrix. We then have :$V\_k(\backslash mathbb\; F^n)\; =\; \backslash left\backslash A\; =\; 1\backslash right\backslash \}.$ The topology on ''V''''k''(F''n'') is the subspace topology inherited from F''n''×''k''. With this topology ''V''''k''(F''n'') is a compact manifold whose dimension is given by :$\backslash dim\; V\_k(\backslash mathbb\; R^n)\; =\; nk\; -\; \backslash frack(k+1)$ :$\backslash dim\; V\_k(\backslash mathbb\; C^n)\; =\; 2nk\; -\; k^2$ :$\backslash dim\; V\_k(\backslash mathbb\; H^n)\; =\; 4nk\; -\; k(2k-1).$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stiefel manifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.