翻訳と辞書 Words near each other ・ Binodal ・ Binodanand Jha ・ Binodini Dasi ・ Binodini Girls' High School ・ Binokor Tashkent ・ Binola, West Virginia ・ Binoli ・ Binomial ・ Binomial (polynomial) ・ Binomial approximation ・ Binomial coefficient ・ Binomial differential equation ・ Binomial distribution ・ Binomial heap ・ Binomial identity ・ Binomial inverse theorem ・ Binomial nomenclature ・ Binomial number ・ Binomial options pricing model ・ Binomial pair ・ Binomial proportion confidence interval ・ Binomial QMF ・ Binomial regression ・ Binomial ring ・ Binomial series ・ Binomial sum variance inequality ・ Binomial test ・ Binomial theorem ・ Binomial transform ・ Binomial type Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Binomial inverse theorem ： ウィキペディア英語版 Binomial inverse theorem In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways. If A, U, B, V are matrices of sizes ''p''×''p'', ''p''×''q'', ''q''×''q'', ''q''×''p'', respectively, then :$$ \left(\mathbf+\mathbf\right)^= \mathbf^ - \mathbf^\mathbf\left(\mathbf+\mathbf^\mathbf\right)^\mathbf^ provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals and the rank of the latter cannot exceed the rank of B.〔Henderson, H. V., and Searle, S. R. (1981), "On deriving the inverse of a sum of matrices", ''SIAM Review'' 23, pp. 53-60.〕 Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with which results in :$$ \left(\mathbf+\mathbf\right)^= \mathbf^ - \mathbf^\mathbf\left(\mathbf^+\mathbf^\mathbf\right)^\mathbf^. This is the Woodbury matrix identity, which can also be derived using matrix blockwise inversion. A more general formula exists when B is singular and possibly even non-square:〔 :$(\backslash mathbf)^=\backslash mathbf^-\backslash mathbf^\backslash mathbf(\backslash mathbf^\backslash mathbf)^\backslash mathbf^.$ Formulas also exist for certain cases in which A is singular.〔Kurt S. Riedel, "A Sherman—Morrison—Woodbury Identity for Rank Augmenting Matrices with Application to Centering", ''SIAM Journal on Matrix Analysis and Applications'', 13 (1992)659-662, (preprint ) 〕 ==Verification== First notice that :$\backslash left(\backslash mathbf\; +\; \backslash mathbf\backslash right)\; \backslash mathbf^\backslash mathbf\; =\; \backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\; =\; \backslash mathbf\; \backslash left(\backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\backslash right).$ Now multiply the matrix we wish to invert by its alleged inverse: :$\backslash left(\backslash mathbf\; +\; \backslash mathbf\backslash right)\; \backslash left(\; \backslash mathbf^\; -\; \backslash mathbf^\backslash mathbf\backslash left(\backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\backslash right)^\backslash mathbf^\; \backslash right)$ :$=\; \backslash mathbf\_p\; +\; \backslash mathbf^\; -\; \backslash mathbf\; \backslash left(\backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\backslash right)\; \backslash left(\backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\backslash right)^\backslash mathbf^$ :$=\; \backslash mathbf\_p\; +\; \backslash mathbf^\; -\; \backslash mathbf^\; =\; \backslash mathbf\_p\; \backslash !$ which verifies that it is the inverse. So we get that if A−1 and $\backslash left(\backslash mathbf\; +\; \backslash mathbf^\backslash mathbf\backslash right)^$ exist, then $\backslash left(\backslash mathbf\; +\; \backslash mathbf\backslash right)^$ exists and is given by the theorem above. 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「Binomial inverse 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.