Editing Tensor (section)

=== Contraction ===
{{Main|Tensor contraction}}

[[Tensor contraction]] is an operation that reduces a type {{nowrap|(''n'', ''m'')}} tensor to a type {{nowrap|(''n'' − 1, ''m'' − 1)}} tensor, of which the [[Trace (linear algebra)|trace]] is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a {{nowrap|(1, 1)}}-tensor <math>T_i^j</math> can be contracted to a scalar through <math>T_i^i</math>, where the summation is again implied.  When the {{nowrap|(1, 1)}}-tensor is interpreted as a linear map, this operation is known as the [[trace (linear algebra)|trace]].

The contraction is often used in conjunction with the tensor product to contract an index from each tensor.

The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space ''V'' with the space ''V''<sup>∗</sup> by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from ''V''<sup>∗</sup> to a factor from ''V''.  For example, a tensor <math>T \in V\otimes V\otimes V^*</math> can be written as a linear combination
:<math>T = v_1\otimes w_1\otimes \alpha_1 + v_2\otimes w_2\otimes \alpha_2 +\cdots + v_N\otimes w_N\otimes \alpha_N.</math>

The contraction of ''T'' on the first and last slots is then the vector
:<math>\alpha_1(v_1)w_1 + \alpha_2(v_2)w_2 + \cdots + \alpha_N(v_N)w_N.</math>

In a vector space with an [[inner product]] (also known as a [[Metric tensor|metric]]) ''g'', the term [[Tensor contraction#Metric contraction|contraction]] is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse.  For example, a {{nowrap|(2, 0)}}-tensor <math>T^{ij} </math> can be contracted to a scalar through <math>T^{ij} g_{ij}</math> (yet again assuming the summation convention).