Tensor (intrinsic definition)

This article assumes an understanding of thetensor productofvector spaceswithout chosenbases.An introduction to the nature and significance of tensors in a broad context can be found in the mainTensorarticle.

Inmathematics,the moderncomponent-freeapproach to the theory of atensorviews a tensor as anabstract object,expressing some definite type ofmultilinearconcept. Their properties can be derived from their definitions, aslinear mapsor more generally; and the rules for manipulations of tensors arise as an extension oflinear algebratomultilinear algebra.

Indifferential geometry,an intrinsicgeometric statement may be described by atensor fieldon amanifold,and then doesn't need to make reference to coordinates at all. The same is true ingeneral relativity,of tensor fields describing aphysical property.The component-free approach is also used extensively inabstract algebraandhomological algebra,where tensors arise naturally.

Definition via tensor products of vector spaces

Given a finite set{V₁,...,V_n}ofvector spacesover a commonfieldF,one may form theirtensor productV₁⊗... ⊗V_n,an element of which is termed atensor.

Atensor on the vector spaceVis then defined to be an element of (i.e., a vector in) a vector space of the form: $$V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}$$ whereV^∗is thedual spaceofV.

If there aremcopies ofVandncopies ofV^∗in our product, the tensor is said to be oftype (m,n)andcontravariantof ordermand covariant of ordernand of totalorderm+n.The tensors of order zero are just the scalars (elements of the fieldF), those of contravariant order 1 are the vectors inV,and those of covariant order 1 are theone-formsinV^∗(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(m,n)is denoted $$T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.$$

Example 1.The space of type(1, 1)tensors,$T_{1}^{1}(V)=V\otimes V^{*},$isisomorphicin a natural way to the space oflinear transformationsfromVtoV.

Example 2.Abilinear formon a real vector spaceV,$V\times V\to F,$corresponds in a natural way to a type(0, 2)tensor in$T_{2}^{0}(V)=V^{*}\otimes V^{*}.$An example of such a bilinear form may be defined,termed the associatedmetric tensor,and is usually denotedg.

Tensor rank

Main article:Tensor rank decomposition

Asimple 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 $$T=a\otimes b\otimes \cdots \otimes d$$ wherea,b,...,dare nonzero and inVorV^∗– that is, if the tensor is nonzero and completelyfactorizable.Every tensor can be expressed as a sum of simple tensors. Therank of a tensorTis the minimum number of simple tensors that sum toT.

Thezero tensorhas 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 isdⁿ⁻¹when each product is ofnvectors from a finite-dimensional vector space of dimensiond.

The termrank of a tensorextends the notion of therank of a matrixin 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 therange of the matrix.A matrix thus has rank one if it can be written as anouter productof two nonzero vectors: $$A=vw^{\mathrm {T} }.$$

The rank of a matrixAis the smallest number of such outer products that can be summed to produce it: $$A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.$$

In indices, a tensor of rank 1 is a tensor of the form $$T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots.$$

The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as amatrix,and can be determined fromGaussian eliminationfor instance. The rank of an order 3 or higher tensor is however oftenvery difficultto 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 anyfinite fieldisNP-Complete,and over the rationals, isNP-Hard.Computational tasks such as the efficient multiplication of matrices and the efficient evaluation ofpolynomialscan be recast as the problem of simultaneously evaluating a set ofbilinear forms $$z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}$$ for given inputsx_iandy_j.If a low-rank decomposition of the tensorTis known, then an efficientevaluation strategyis known.

Universal property

The space$T_{n}^{m}(V)$can be characterized by auniversal propertyin terms ofmultilinear 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 forfree modules,and the "universal" approach carries over more easily to more general situations.

A scalar-valued function on aCartesian product(ordirect sum) of vector spaces $$f:V_{1}\times \cdots \times V_{N}\to F$$ is multilinear if it is linear in each argument. The space of all multilinear mappings fromV₁×... ×V_NtoWis denotedL^N(V₁,...,V_N;W).WhenN= 1,a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings fromVtoWis denotedL(V;W).

Theuniversal characterization of the tensor productimplies that, for each multilinear function $$f\in L^{m+n}(\underbrace {V^{*},\ldots,V^{*}} _{m},\underbrace {V,\ldots,V} _{n};W)$$ (whereWcan represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function $$T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)$$ such that $$f(\alpha _{1},\ldots,\alpha _{m},v_{1},\ldots,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})$$ for allv_iinVandα_iinV^∗.

Using the universal property, it follows, whenVisfinite dimensional,that the space of(m,n)-tensors admits anatural isomorphism $$T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots,V^{*}} _{m},\underbrace {V,\ldots,V} _{n};F).$$

EachVin the definition of the tensor corresponds to aV^∗inside the argument of the linear maps, and vice versa. (Note that in the former case, there aremcopies ofVandncopies ofV^∗,and in the latter case vice versa). In particular, one has $${\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V,\\T_{1}^{0}(V)&\cong L(V;F)=V^{*},\\T_{1}^{1}(V)&\cong L(V;V).\end{aligned}}$$

Tensor fields

Main article:tensor field

Differential geometry,physicsandengineeringmust often deal withtensor fieldsonsmooth manifolds.The termtensoris sometimes used as a shorthand fortensor field.A tensor field expresses the concept of a tensor that varies from point to point on the manifold.

References

• Abraham, Ralph;Marsden, Jerrold E.(1985),Foundations of Mechanics(2nd ed.), Reading, Massachusetts: Addison-Wesley,ISBN0-201-40840-6.
• Bourbaki, Nicolas(1989),Elements of Mathematics, Algebra I,Springer-Verlag,ISBN3-540-64243-9.
• de Groote, H. F. (1987),Lectures on the Complexity of Bilinear Problems,Lecture Notes in Computer Science, vol. 245, Springer,ISBN3-540-17205-X.
• Halmos, Paul(1974),Finite-dimensional Vector Spaces,Springer,ISBN0-387-90093-4.
• Håstad, Johan (November 15, 1989),"Tensor Rank Is NP-Complete",Journal of Algorithms,11:644–654.
• Jeevanjee, Nadir (2011), "An Introduction to Tensors and Group Theory for Physicists",Physics Today,65(4): 64,Bibcode:2012PhT....65d..64P,doi:10.1063/PT.3.1523,ISBN978-0-8176-4714-8.
• Knuth, Donald E.(1998) [1969],The Art of Computer Programming,vol. 2 (3rd ed.), pp. 145–146,ISBN978-0-201-89684-8.
• Hackbusch, Wolfgang(2012),Tensor Spaces and Numerical Tensor Calculus,Springer, p. 4,ISBN978-3-642-28027-6.