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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 Tensor contraction ： ウィキペディア英語版 Tensor contraction In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In the Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. Tensor contraction can be seen as a generalization of the trace. == Abstract formulation == Let ''V'' be a vector space over a field ''k''. The core of the contraction operation, and the simplest case, is the natural pairing of ''V'' with its dual vector space ''V'' *. The pairing is the linear transformation from the tensor product of these two spaces to the field ''k'': : $C\; :\; V^$ * \otimes V \rightarrow k corresponding to the bilinear form : $\backslash langle\; f,\; v\; \backslash rangle\; =\; f(v)$ where ''f'' is in ''V'' * and ''v'' is in ''V''. The map ''C'' defines the contraction operation on a tensor of type (1,1), which is an element of $V^$ * \otimes V . Note that the result is a scalar (an element of ''k''). Using the natural isomorphism between $V^$ * \otimes V and the space of linear transformations from ''V'' to ''V'',〔Let L(''V'',''V'') be the space of linear transformations from ''V'' to ''V''. Then the natural map : $V^$ * \otimes V \rightarrow L(V,V) is defined by : $f\; \backslash otimes\; v\; \backslash mapsto\; g$ where ''g''(''w'') = ''f''(''w'')''v''. Suppose that ''V'' is finite-dimensional. If is a basis of ''V'' and is the corresponding dual basis, then $f^i\; \backslash otimes\; v\_j$ maps to the transformation whose matrix in this basis has only one nonzero entry, a 1 in the ''i'',''j'' position. This shows that the map is an isomorphism.〕 one obtains a basis-free definition of the trace. In general, a tensor of type (''m'', ''n'') (with ''m'' ≥ 1 and ''n'' ≥ 1) is an element of the vector space : $V\; \backslash otimes\; \backslash cdots\; \backslash otimes\; V\; \backslash otimes\; V^\; \backslash otimes\; \backslash cdots\; \backslash otimes\; V^$ (where there are ''m'' ''V'' factors and ''n'' ''V *'' factors). Applying the natural pairing to the ''k''th ''V'' factor and the ''l''th ''V *'' factor, and using the identity on all other factors, defines the (''k'', ''l'') contraction operation, which is a linear map which yields a tensor of type (''m'' − 1, ''n'' − 1).〔 By analogy with the (1,1) case, the general contraction operation is sometimes called the trace. 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「Tensor contraction」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.