Tensor (intrinsic definition)

For an introduction to the nature and significance of tensors in a broad context, see Tensor.

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.

In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally.

Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.

Definition via tensor products of vector spaces

Given a finite set { V₁, ..., V_n } of vector spaces over a common field F, one may form their tensor product V₁ ⊗ ... ⊗ V_n, an element of which is termed a tensor.

A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:

$V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*$

where V* is the dual space of V.

If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m+n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m,n) is denoted

$T^m_n(V) = \underbrace{ V\otimes \dots \otimes V}_{m} \otimes \underbrace{ V^*\otimes \dots \otimes V^*}_{n} .$

The (1,1) tensors

$V \otimes V^*$

are isomorphic in a natural way to the space of linear transformations from V to V. A bilinear form on a real vector space V; V × V → R corresponds in a natural way to a (0,2) tensor in

$V^* \otimes V^*$

termed the associated metric tensor (or sometimes misleadingly the metric or inner product) and usually denoted g.

Tensor rank

Main article: Tensor rank decomposition

The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:

$A = v w^{\mathrm{T}}.$

More generally, the rank of a matrix A is the smallest number of such outer products that can be summed to produce it:

$A = v_1w_1^\mathrm{T} + \cdots + v_kw_k^\mathrm{T}.$

Similarly, a tensor of rank one (also called a simple tensor) is a tensor that can be written as a tensor product of the form

$T=a\otimes b\otimes\cdots\otimes d$

where a, b, ..., d are nonzero and in V or V*. That is, if the tensor is nonzero and completely factorizable. In indices, a tensor of rank 1 is a tensor of the form

$T_{ij\dots}^{k\ell\dots}=a_ib_j\cdots c^kd^\ell\cdots.$

Every tensor can be expressed as a sum of rank 1 tensors. The rank of a general tensor T is defined to be the minimum number of rank 1 tensors with which it is possible to express T as a sum (Bourbaki 1989, II, §7, no. 8).

A nonzero order 1 tensor always has rank 1. The zero tensor has rank zero. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of all but the longest of the lengths of the vectors in (a sum of products of) which the tensor can be expressed. The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms

$z_k = \sum_{ij} T_{ijk}x_iy_j\,$

for given inputs x_i and y_j. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known (Knuth 1998, pp. 506–508).

Universal property

The space $T^m_n(V)$ can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.

A scalar-valued function on a Cartesian product (or direct sum) of vector spaces

$f : V_1\times V_2\times\cdots\times V_N \to \mathbf{R}$

is multilinear if it is linear in each argument. The space of all multlinear mappings from the product V₁×V₂×...×V_N into W is denoted L^N(V₁,V₂,...,V_N; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V;W).

The universal characterization of the tensor product implies that, for each multilinear function

$f\in L^{m+n}(\underbrace{V,V,\dots,V}_m,\underbrace{V^*,V^*,\dots,V^*}_n;W)$

there exists a unique linear function

$T_f \in L(V\otimes\cdots\otimes V\otimes V^*\otimes\cdots\otimes V^*; W)$

such that

$f(v_1,\dots,v_m,\alpha_1,\dots,\alpha_n) = T_f(v_1\otimes\cdots\otimes v_m\otimes\alpha_1\otimes\cdots\otimes\alpha_n)$

for all v_i ∈ V and α_i ∈ V^∗.

Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism

$T^m_n(V) \cong L(V^* \otimes \dots \otimes V^* \otimes V \otimes \dots \otimes V ; \mathbb{R}) \cong L^{m+n}(V^*,\dots,V^*,V,\dots,V;\mathbb{R}).$

In the formula above, the roles of V and V^* are reversed. In particular, one has

$T^1_0(V) \cong L(V^*;\mathbb{R}) \cong V$

and

$T^0_1(V) \cong L(V;\mathbb{R}) = V^*$

and

$T^1_1(V) \cong L(V;V).$

Tensor fields

Main article: tensor field

Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point.

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