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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 Lagrange's identity ： ウィキペディア英語版 Lagrange's identity In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:〔 〕〔 〕 :$$ \begin \biggl( \sum_^n a_k^2\biggr) \biggl(\sum_^n b_k^2\biggr) - \biggl(\sum_^n a_k b_k\biggr)^2 & = \sum_^ \sum_^n (a_i b_j - a_j b_i)^2 \\ & \biggl(= \frac \sum_^n \sum_^n (a_i b_j - a_j b_i)^2\biggr), \end which applies to any two sets and of real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta-Fibonacci identity and a special form of the Binet–Cauchy identity. In a more compact vector notation, Lagrange's identity is expressed as:〔 〕 :$\backslash |\; \backslash mathbf\; a\; \backslash |^2\; \backslash \; \backslash |\; \backslash mathbf\; b\; \backslash |^2\; -\; (\backslash mathbf\; )^2\; =\; \backslash sum\_\; \backslash left(a\_ib\_j-a\_jb\_i\; \backslash right)^2\; \backslash \; ,$ where a and b are ''n''-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form:〔 〕 :$\backslash biggl(\; \backslash sum\_^n\; |a\_k|^2\backslash biggr)\; \backslash biggl(\backslash sum\_^n\; |b\_k|^2\backslash biggr)\; -\; \backslash biggl|\backslash sum\_^n\; a\_k\; b\_k\backslash biggr|^2\; =\; \backslash sum\_^\; \backslash sum\_^n\; |a\_i\; \backslash overline\_j\; -\; a\_j\; \backslash overline\_i|^2$ involving the absolute value.〔 ; . 〕 Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space ℝ''n'' and its complex counterpart ℂ''n''. Geometrically, the identity asserts that the square of the volume of the parallelepiped spanned by a set of vectors is the Gram determinant of the vectors. ==Lagrange's identity and exterior algebra== In terms of the wedge product, Lagrange's identity can be written :$(a\; \backslash cdot\; a)(b\; \backslash cdot\; b)\; -\; (a\; \backslash cdot\; b)^2\; =\; (a\; \backslash wedge\; b)\; \backslash cdot\; (a\; \backslash wedge\; b).$ Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the parallelogram they define, in terms of the dot products of the two vectors, as :$\backslash |a\; \backslash wedge\; b\backslash |\; =\; \backslash sqrt.$ 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「Lagrange's 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.