Tensor Contraction - Contraction of A Pair of Tensors

Contraction of A Pair of Tensors

One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensors T and U. The tensor product is a new tensor, which, if it has at least one covariant and one contravariant index, can be contracted. The case where T is a vector and U is a dual vector is exactly the core operation introduced first in this article.

In abstract index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.

For example, matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant. Let be the components of one matrix and let be the components of a second matrix. Then their multiplication is given by the following contraction, an example of the contraction of a pair of tensors:

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Also, the interior product of a vector with a differential form is a special case of the contraction of two tensors with each other.