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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 Sylvester's determinant theorem ： ウィキペディア英語版 Sylvester's determinant theorem In matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this theorem without proof in 1851.〔 Cited in 〕 The theorem states that if ''A'', ''B'' are matrices of size ''p'' × ''n'' and ''n'' × ''p'' respectively, then :$\backslash det(I\_p\; +\; AB)\; =\; \backslash det(I\_n\; +\; BA),\backslash$ where ''I''''a'' is the identity matrix of order ''a''.〔 page 416〕 This can be seen for invertible ''A'', ''B'' by conjugating ''I + AB'' by ''A−1'', then extended to arbitrary square matrices by density of invertible matrices, and then to arbitrary rectangular matrices by adding zero column or row vectors as necessary. It is closely related to the Matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses. ==Proof== The theorem may be proven as follows.〔.〕 Let $M$ be a matrix comprising the four blocks $-A$, $B$, $I\_n$ and $I\_p$ :. Block LU decomposition of $M$ yields : from which :$\backslash det(M)\; =\; \backslash det(I\_n\; +\; B\; A)$ follows. Decomposing $M$ to an upper and a lower triangular matrix instead, :, yields :$\backslash det(M)\; =\; \backslash det(I\_p\; +\; A\; B)$. This proves :$\backslash det(I\_n\; +\; B\; A)\; =\; \backslash det(I\_p\; +\; A\; B)$. 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「Sylvester's determinant theorem」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.