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 multilinear concept. Their 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 extensively 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.
Given a finite set of vector spaces over a common field F, one may form their tensor product, 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 ⊗ … ⊗ V ⊗ V* ⊗ … ⊗ 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 and contravariant of order m and covariant of order n and of total order . 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 elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type is denoted
| m | |
| T | |
| n(V) |
=\underbrace{V ⊗ ... ⊗ V}m ⊗ \underbrace{V* ⊗ ... ⊗
| *} | |
| V | |
| n |
.
Example 1. The space of type tensors,
| 1 | |
| T | |
| 1(V) |
=V ⊗ V*,
Example 2. A bilinear form on a real vector space V,
V x V\toF,
| 0 | |
| T | |
| 2 |
(V)=V* ⊗ V*.
See main article: Tensor rank decomposition.
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can be written as a product of tensors of the form
T=a ⊗ b ⊗ … ⊗ d
where a, b, ..., d are nonzero and in V or V∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the minimum number of simple tensors that sum to T .
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is d when each product is of n vectors from a finite-dimensional vector space of dimension d.
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=vwT.
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
A=v1w
| T | |
| 1 |
+ … +vk
| T. | |
| w | |
| k |
In indices, a tensor of rank 1 is a tensor of the form
| k\ell... | |
| T | |
| ij... |
=aibj … ckd\ell … .
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix, 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 . In fact, the problem of finding the rank of an order 3 tensor over any finite field is NP-Complete, and over the rationals, is NP-Hard.[1] 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
zk=\sumijTijkxiyj
for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known .
The space
| m | |
| T | |
| n(V) |
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
f:V1 x … x VN\toF
is multilinear if it is linear in each argument. The space of all multilinear mappings from to W is denoted LN(V1, ..., VN; 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 .
The universal characterization of the tensor product implies that, for each multilinear function
f\inLm+n(\underbrace{V*,\ldots,V
| *} | |
| m,\underbrace{V,\ldots,V} |
n;W)
(where
W
Tf\inL(\underbrace{V* ⊗ … ⊗
| *} | |
| V | |
| m |
⊗ \underbrace{V ⊗ … ⊗ V}n;W)
such that
f(\alpha1,\ldots,\alpham,v1,\ldots,vn)=Tf(\alpha1 ⊗ … ⊗ \alpham ⊗ v1 ⊗ … ⊗ vn)
for all
vi\inV
\alphai\inV*.
Using the universal property, it follows, when V is finite dimensional, that the space of (m,n)-tensors admits a natural isomorphism
| m | |
| T | |
| n(V) |
\congL(\underbrace{V* ⊗ … ⊗
| *} | |
| V | |
| m |
⊗ \underbrace{V ⊗ … ⊗ V}n;F)\congLm+n(\underbrace{V*,
| *} | |
| \ldots,V | |
| m,\underbrace{V,\ldots,V} |
n;F).
Each V in the definition of the tensor corresponds to a V* inside the argument of the linear maps, and vice versa. (Note that in the former case, there are m copies of V and n copies of V*, and in the latter case vice versa). In particular, one has
| 1 | |
| \begin{align} T | |
| 0(V) |
&\congL(V*;F)\cong
| 0 | |
| V,\\ T | |
| 1(V) |
&\congL(V;F)=V*,\\ T
| 1 | |
| 1(V) |
&\congL(V;V). \end{align}
See 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 on the manifold.