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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 Nilpotent matrix ： ウィキペディア英語版 Nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :$N^k\; =\; 0\backslash ,$ for some positive integer ''k''. The smallest such ''k'' is sometimes called the degree or index of ''N''. More generally, a nilpotent transformation is a linear transformation ''L'' of a vector space such that ''L''''k'' = 0 for some positive integer ''k'' (and thus, ''L''''j'' = 0 for all ''j'' ≥ ''k''). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings. ==Examples== The matrix :$$ M = \begin 0 & 1 \\ 0 & 0 \end is nilpotent, since ''M''2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix :$$ N = \begin 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end is nilpotent, with :$$ N^2 = \begin 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end ;\ N^3 = \begin 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end ;\ N^4 = \begin 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end. Though the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, the matrix :$$ N = \begin 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end squares to zero, though the matrix has no zero entries. 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「Nilpotent matrix」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.