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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 Matrix of ones ： ウィキペディア英語版 Matrix of ones In mathematics, a matrix of ones or all-ones matrix is a matrix where every element is equal to one.〔.〕 Examples of standard notation are given below: :$J\_2=\backslash begin$ 1 & 1 \\ 1 & 1 \end;\quad J_3=\begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end;\quad J_=\begin 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end.\quad Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different matrix. ==Properties== For an ''n×n'' matrix of ones ''J'', the following properties hold: * The trace of ''J'' is ''n'',〔.〕 and the determinant is 1 if ''n'' is 1, or 0 otherwise. * The rank of ''J'' is 1 and the eigenvalues are ''n'' (once) and 0 (''n''-1 times).〔; , (p. 65 ).〕 * ''J'' is positive semi-definite matrix. This follows from the previous property. *$J^k\; =\; n^\; J,\; \backslash mbox\; k=1,2,\backslash ldots.\backslash ,$〔.〕 *The matrix $\backslash tfrac1n\; J$ is idempotent. This is a simple corollary of the above.〔 *$\backslash exp(J)\; =\; I\; +\; \backslash frac\; J,$ where exp(''J'') is the matrix exponential. * ''J'' is the neutral element of the Hadamard product.〔.〕 *If ''A'' is the adjacency matrix of a ''n''-vertex undirected graph ''G'', and ''J'' is the all-ones matrix of the same dimension, then ''G'' is a regular graph if and only if ''AJ'' = ''JA''.〔.〕 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「Matrix of ones」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.