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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 Woodbury matrix identity ： ウィキペディア英語版 Woodbury matrix identity In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury〔Max A. Woodbury, ''Inverting modified matrices'', Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950, 4pp 〕〔Max A. Woodbury, ''The Stability of Out-Input Matrices''. Chicago, Ill., 1949. 5 pp. 〕 says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the original matrix. Alternative names for this formula are the ''matrix inversion lemma'', Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is〔 〕 :$\backslash left(A+UCV\; \backslash right)^\; =\; A^\; -\; A^U\; \backslash left(C^+VA^U\; \backslash right)^\; VA^,$ where ''A'', ''U'', ''C'' and ''V'' all denote matrices of the correct (conformable) sizes. Specifically, ''A'' is ''n''-by-''n'', ''U'' is ''n''-by-''k'', ''C'' is ''k''-by-''k'' and ''V'' is ''k''-by-''n''. This can be derived using blockwise matrix inversion. For a more general formula for which the matrix ''C'' need not be invertible or even square, see Binomial inverse theorem. In the special case where ''C'' is the 1-by-1 unit matrix, this identity reduces to the Sherman–Morrison formula. In the special case when ''C'' is the identity matrix ''I'', the matrix $I+VA^U$ is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.〔 == Direct proof == The formula can be proven by checking that $(A+UCV)$ times its alleged inverse on the right side of the Woodbury identity gives the identity matrix: : $$ \begin &\left(A+UCV \right) \left(A^ - A^U \left(C^+VA^U \right)^ VA^ \right ) \\ & \quad = I + UCVA^ - (U+UCVA^U)(C^ + VA^U)^VA^ \\ & \quad = I + UCVA^ - UC(C^+ VA^U)(C^ + VA^U)^VA^ \\ & \quad = I + UCVA^ - UCVA^ = I. \end 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「Woodbury matrix identity」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.