翻訳と辞書 Words near each other ・ Grammia brillians ・ Grammia cervinoides ・ Grammia complicata ・ Grammia doris ・ Grammia edwardsii ・ Grammia elongata ・ Grammia eureka ・ Grammia f-pallida ・ Grammia favorita ・ Grammia fergusoni ・ Grammia figurata ・ Grammia franconia ・ Gramercy Theatre ・ Gramercy, Louisiana ・ Gramgeeta ・ Gramian matrix ・ Gramiccia Lorenzo ・ Gramicidin ・ Gramicidin B ・ Gramicidin S ・ Gramilla, Santiago del Estero ・ Gramimond ・ Gramin Bank of Aryavart ・ Gramin Gyan Library ・ Gramin Vikas Vigyan Samiti ・ Gramine ・ Graminea ・ Graminea annulata ・ Graminea hispida ・ Graminea inca 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 Gramian matrix ： ウィキペディア英語版 Gramian matrix In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors $v\_1,\backslash dots,\; v\_n$ in an inner product space is the Hermitian matrix of inner products, whose entries are given by $G\_=\backslash langle\; v\_i,\; v\_j\; \backslash rangle$. For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply $G\; =\; V^\backslash mathrm\; V$ (or $G\; =\; V^\backslash dagger\; V$ for complex vectors using the conjugate transpose), where ''V'' is a matrix whose columns are the vectors $v\_k$. An important application is to compute linear independence: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. ==Examples== Most commonly, the vectors are elements of a Euclidean space, or are functions in an ''L''2 space, such as continuous functions on a compact interval () (which are a subspace of ''L'' 2(())). Given real-valued functions $\backslash$ on the interval $()$, the Gram matrix $G=()$, is given by the standard inner product on functions: : $G\_=\backslash int\_^\; \backslash ell\_i(\backslash tau)\backslash bar(\backslash tau)\backslash ,\; d\backslash tau.$ Given a real matrix ''A'', the matrix ''A''T''A'' is a Gram matrix (of the columns of ''A''), while the matrix ''AA''T is the Gram matrix of the rows of ''A''. For a general bilinear form ''B'' on a finite-dimensional vector space over any field we can define a Gram matrix ''G'' attached to a set of vectors $v\_1,\backslash dots,\; v\_n$ by $G\_\; =\; B(v\_i,v\_j)\; \backslash ,$. The matrix will be symmetric if the bilinear form ''B'' is symmetric. 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「Gramian matrix」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.