Editing Tensor (section)

== Operations ==
There are several operations on tensors that again produce a tensor.  The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the [[Scalar multiplication|scaling of a vector]].  On components, these operations are simply performed component-wise.  These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type.

=== Tensor product ===
{{Main|Tensor product}}

The [[tensor product]] takes two tensors, ''S'' and ''T'', and produces a new tensor, {{nowrap|{{math|''S'' ⊗ ''T''}}}}, whose order is the sum of the orders of the original tensors.  When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.,
<math display="block">(S \otimes T)(v_1, \ldots, v_n, v_{n+1}, \ldots, v_{n+m}) = S(v_1, \ldots, v_n)T(v_{n+1}, \ldots, v_{n+m}),</math>
which again produces a map that is linear in all its arguments.  On components, the effect is to multiply the components of the two input tensors pairwise, i.e.,
<math display="block">
  (S \otimes T)^{i_1\ldots i_l i_{l+1}\ldots i_{l+n}}_{j_1\ldots j_k j_{k+1}\ldots j_{k+m}} =
  S^{i_1\ldots i_l}_{j_1\ldots j_k} T^{i_{l+1}\ldots i_{l+n}}_{j_{k+1}\ldots j_{k+m}}.
</math>
If {{mvar|S}} is of type {{math|(''l'', ''k'')}} and {{mvar|T}} is of type {{math|(''n'', ''m'')}}, then the tensor product {{nowrap|{{math|''S'' ⊗ ''T''}}}} has type {{nowrap|{{math|(''l'' + ''n'', ''k'' + ''m'')}}}}.

=== 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).

=== Raising or lowering an index ===
{{Main|Raising and lowering indices}}
When a vector space is equipped with a [[nondegenerate bilinear form]] (or ''[[metric tensor]]'' as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa.  A metric tensor is a (symmetric) ({{nowrap|0, 2)}}-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product.  This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.  This operation is quite graphically known as ''lowering an index''.

Conversely, the inverse operation can be defined, and is called ''raising an index''.  This is equivalent to a similar contraction on the product with a {{nowrap|(2, 0)}}-tensor.  This ''inverse metric tensor'' has components that are the matrix inverse of those of the metric tensor.