Tensors

Tensor contraction

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. Tensor contraction can be seen as a generalization of the trace. (Wikipedia).

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Related pages

Covariant derivative | Topological space | Vector space | Differential form | Tensor product | Trace (linear algebra) | Analytic space | Musical isomorphism | Complex manifold | Mixed tensor | Dimension | Divergence | Riemann curvature tensor | Partial trace | Continuity equation | Interior product | Natural transformation | Dual space | Field (mathematics) | Riemannian manifold | Sheaf (mathematics) | Euclidean space | Tensor | Scalar (mathematics) | Bilinear form | Manifold | Ricci calculus | Scheme (mathematics) | Metric tensor | Einstein notation | Covariance and contravariance of vectors | Multilinear algebra | Vector field | Scalar curvature | Module (mathematics) | Raising and lowering indices | Commutative ring