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In multilinear algebra, a tensor contraction is an operation on a tensor 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 that 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 Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2.

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rdf:type	Thing yago:WikicatTensors yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Quantity105855125 military conflict yago:Tensor105864481 yago:Variable105857459
rdfs:label	Tensorverjüngung (de) Contraction tensorielle (fr) Contrazione di un tensore (it) テンソルの縮約 (ja) Tensor contraction (en) Свёртка тензора (ru) Згортка тензора (uk)
rdfs:comment	Die Tensorverjüngung oder Kontraktion ist ein mathematischer Begriff aus der linearen Algebra mit Verwendung in der Tensoranalysis und Tensoralgebra. Es ist eine Verallgemeinerung der Spur einer linearen Abbildung auf Tensoren, die mindestens einfach kovariant und einfach kontravariant sind. Anwendungen finden sich z. B. in der Relativitätstheorie (siehe auch Längenkontraktion), Mechanik usw. (de) En algèbre multilinéaire, la contraction est un procédé de calcul sur les tenseurs faisant intervenir la dualité. En coordonnées elle se représente de façon très simple en utilisant les notations d'Einstein et consiste à faire une somme sur un indice muet. Il est possible de contracter un tenseur unique de rang p en un tenseur de rang p-2, par exemple en calculant la trace d'une matrice. Il est possible également de contracter deux tenseurs, ce qui généralise la notion de produit matriciel. (fr) 多重線型代数学におけるテンソルの縮約（テンソルのしゅくやく、英: tensor contraction）は、有限次元のベクトル空間とその双対空間の間の自然な内積から生じる、一つ以上のテンソルに対する演算である。座標を取って考えれば、一つの式に現れる各々の仮添字 (dummy index) の対に対して和の規約を適用することによって生じる、スカラー成分の積和として縮約は表される。特に一つのの縮約は、そのテンソルに現れる見かけの添字の対（一方は上付き、他方は下付き）が同じ文字であるとき、それらに関して和をとることで生じる。アインシュタインの縮約記法とは、このような和を織り込み済みとする記法である。縮約を取って得られるテンソルは階数 (order) が 2 だけ減る。テンソルの縮約をトレースの一般化として捉えることもできる。 (ja) In geometria differenziale, la contrazione di un tensore è un'operazione che trasforma un tensore di tipo in un tensore di tipo . Questa operazione è a volte detta traccia. Se il tensore è di tipo (1,1), questa equivale effettivamente al calcolo della traccia di una matrice associata. (it) Свёртка в тензорном исчислении — операция понижения валентности тензора на 2, переводящая тензор валентности в тензор валентности . (ru) In multilinear algebra, a tensor contraction is an operation on a tensor 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 that 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 Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. (en) Згортка в тензорному численні — операція пониження валентності тензора на 2, котра переводить тензор валентності в тензор валентності . Згортку можна розглядати як узагальнення сліду матриці. В координатах вона записується таким чином: де застосовано правило сумування Ейнштейна за різноваріантними індексами, що повторюються. Часто операцію згортки проводять над тензорами, що є добутками тензорів. Наприклад, є запис звичайного множення матриці А на матрицю B (тобто ). (uk)
rdfs:seeAlso	Lowering indices Raising
dcterms:subject	Tensors
Wikipage page ID	202722 (xsd:integer)
Wikipage revision ID	1124792159 (xsd:integer)
Link from a Wikipage to another Wikipage	Cartesian coordinates Scalar (mathematics) Scalar curvature Module (mathematics) Partial trace Bilinear form Ricci calculus Vector field Vector space Interior product Continuity equation Covariance and contravariance of vectors Analytic space Generalization Natural transformation Einstein notation Multilinear algebra Musical isomorphism Commutative ring Complex manifold Trace (linear algebra) Dual space Euclidean space Field (mathematics) Riemann curvature tensor Riemannian manifold Covariant derivative Tensor Tensor fields Tensor product Tensors Mixed tensor Differential form Dimension Divergence Manifold Tensor index notation Inner product Metric tensor Raising and lowering indices Scheme (mathematics) Sheaf (mathematics) Linear transformation Metric (mathematics) Topological space Dual vector space Dyadic tensor Covariant differential Natural pairing Ricci tensor Summation convention Component-free treatment of tensors Locally free sheaf Structure sheaf dbr:Metric_contraction
Link from a Wikipage to an external page	https://archive.org/details/tensoranalysison00bish
sameAs	Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction Tensor contraction
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