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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 Poincaré separation theorem ： ウィキペディア英語版 Poincaré separation theorem In mathematics, the Poincaré separation theorem gives the upper and lower bounds of eigenvalues of a real symmetric matrix ''B'AB'' that can be considered as the orthogonal projection of a larger real symmetric matrix ''A'' onto a linear subspace spanned by the columns of ''B''. The theorem is named after Henri Poincaré. More specifically, let ''A'' be an ''n'' × ''n'' real symmetric matrix and ''B'' an ''n'' × ''r'' semi-orthogonal matrix such that ''B'B'' = ''I''''r''. Denote by $\backslash lambda\_i$, ''i'' = 1, 2, ..., ''n'' and $\backslash mu\_i$, ''i'' = 1, 2, ..., ''r'' the eigenvalues of ''A'' and ''B'AB'', respectively (in descending order). We have : $\backslash lambda\_i\; \backslash geq\; \backslash mu\_i\; \backslash geq\; \backslash lambda\_,$ == Proof == An algebraic proof, based on the variational interpretation of eigenvalues, has been published in Magnus' ''Matrix Differential Calculus with Applications in Statistics and Econometrics''. From the geometric point of view, ''B'AB'' can be considered as the orthogonal projection of ''A'' onto the linear subspace spanned by ''B'', so the above results follow immediately. 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「Poincaré separation theorem」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.