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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 Euclidean distance matrix ： ウィキペディア英語版 Euclidean distance matrix In mathematics, a Euclidean distance matrix is an ''n×n'' matrix representing the spacing of a set of ''n'' points in Euclidean space. If ''A'' is a Euclidean distance matrix and the points $x\_1,x\_2,\backslash ldots,x\_n$ are defined on ''m''-dimensional space, then the elements of ''A'' are given by :$\backslash begin$ A & = & (a_); \\ a_ & = & ||x_i - x_j||_2^2 \end where ||.||2 denotes the 2-norm on Rm. ==Properties== Simply put, the element $a\_$ describes the square of the distance between the ''i'' th and ''j'' th points in the set. By the properties of the 2-norm (or indeed, Euclidean distance in general), the matrix ''A'' has the following properties. * All elements on the diagonal of ''A'' are zero (i.e. it is a hollow matrix). * The trace of ''A'' is zero (by the above property). * ''A'' is symmetric (i.e. $a\_\; =\; a\_$). * $\backslash sqrt\}\; +\; \backslash sqrt\backslash ge\; 0$ * The number of unique (distinct) non-zero values within an ''n''-by-''n'' Euclidean distance matrix is bounded above by $n(n-1)/2$ due to the matrix being symmetric and hollow. * In dimension ''m'', a Euclidean distance matrix has rank less than or equal to ''m+2''. If the points $x\_1,x\_2,\backslash ldots,x\_n$ are in general position, the rank is exactly 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「Euclidean distance 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.