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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 Cauchy–Binet formula ： ウィキペディア英語版 Cauchy–Binet formula In linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with entries from any commutative ring. == Statement == Let ''A'' be an ''m''×''n'' matrix and ''B'' an ''n''×''m'' matrix. Write () for the set , and $\backslash tbinomm$ for the set of ''m''-combinations of () (i.e., subsets of size ''m''; there are $\backslash tbinom\; nm$ of them). For $S\backslash in\backslash tbinomm$, write ''A''(),''S'' for the ''m''×''m'' matrix whose columns are the columns of ''A'' at indices from ''S'', and ''B''''S'',() for the ''m''×''m'' matrix whose rows are the rows of ''B'' at indices from ''S''. The Cauchy–Binet formula then states : $\backslash det(AB)\; =\; \backslash sum\_\; \backslash det(A\_)\backslash det(B\_).$ Example: taking ''m'' = 2 and ''n'' = 3, and matrices 3& 1& -1\\ \end and $B\; =$ \begin1&1\\3&1\\0&2\end, the Cauchy–Binet formula gives the determinant: :$$ \det(AB)= \left|\begin1&1\\3&1\end\right| \cdot \left|\begin1&1\\3&1\end\right| + \left|\begin1&2\\1&-1\end\right| \cdot \left|\begin3&1\\0&2\end\right| + \left|\begin1&2\\3&-1\end\right| \cdot \left|\begin1&1\\0&2\end\right|. Indeed , and its determinant is −28, which is also the value $-2\backslash times-2+-3\backslash times6+-7\backslash times2$ given by the right hand side of the formula. 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「Cauchy–Binet formula」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.