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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 Laplace expansion ： ウィキペディア英語版 Laplace expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |''B''| of an ''n'' × ''n'' matrix ''B'' that is a weighted sum of the determinants of ''n'' sub-matrices of ''B'', each of size (''n''−1) × (''n''−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation. The ''i'', ''j'' ''cofactor'' of ''B'' is the scalar ''Cij'' defined by :$C\_\backslash \; =\; (-1)^\; M\_\backslash ,,$ where ''Mij'' is the ''i'', ''j'' ''minor matrix'' of ''B'', that is, the determinant of the (''n'' − 1) × (''n'' − 1) matrix that results from deleting the ''i''-th row and the ''j''-th column of ''B''. Then the Laplace expansion is given by the following :Theorem. Suppose ''B'' = () is an ''n'' × ''n'' matrix and fix any ''i'', ''j'' ∈ . Then its determinant |''B''| is given by: :$\backslash begin$ |B| & = b_ C_ + b_ C_ + \cdots + b_ C_ \\ & = b_ C_ + b_ C_ + \cdots + b_ C_ \\ & = \sum_^ b_ C_ = \sum_^ b_ C_ \end == Examples == Consider the matrix : The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields: : ::: :::$\{\}\; =\; -2\; \backslash cdot\; (-6)\; +\; 5\; \backslash cdot\; (-12)\; -\; 8\; \backslash cdot\; (-6)\; =\; 0.$ It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero. 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「Laplace expansion」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.