Editing Tensor (section)

== Definition ==
Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.

=== As multidimensional arrays ===
A tensor may be represented as a (potentially multidimensional) array. Just as a [[Vector space|vector]] in an {{mvar|n}}-[[dimension (vector space)|dimensional]] space is represented by a [[multidimensional array|one-dimensional]] array with {{mvar|n}} components with respect to a given [[Basis (linear algebra)#Ordered bases and coordinates|basis]], any tensor with respect to a basis is represented by a multidimensional array.  For example, a [[linear operator]] is represented in a basis as a two-dimensional square {{math|''n'' × ''n''}} array.  The numbers in the multidimensional array are known as the ''components'' of the tensor.  They are denoted by indices giving their position in the array, as [[subscript and superscript|subscripts and superscripts]], following the symbolic name of the tensor.  For example, the components of an order-{{math|2}} tensor {{mvar|T}} could be denoted {{math|''T''<sub>''ij''</sub>}} , where {{mvar|i}} and {{mvar|j}} are indices running from {{math|1}} to {{mvar|n}}, or also by {{math|''T''{{thinsp}}{{su|lh=0.8|b=''j''|p=''i''}}}}.  Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while {{math|''T''<sub>''ij''</sub>}} and {{math|''T''{{thinsp}}{{su|lh=0.8|b=''j''|p=''i''}}}} can both be expressed as ''n''-by-''n'' matrices, and are numerically related via [[Raising and lowering indices|index juggling]], the difference in their transformation laws indicates it would be improper to add them together.

The total number of indices ({{mvar|m}}) required to identify each component uniquely is equal to the ''dimension'' or the number of ''ways'' of an array, which is why a tensor is sometimes referred to as an {{mvar|m}}-dimensional array or an {{mvar|m}}-way array.  The total number of indices is also called the ''order'', ''degree'' or ''rank'' of a tensor,<ref name=DeLathauwerEtAl2000 >{{cite journal| last1= De Lathauwer |first1= Lieven| last2= De Moor |first2= Bart| last3= Vandewalle |first3= Joos| date=2000|title=A Multilinear Singular Value Decomposition |journal= [[SIAM J. Matrix Anal. Appl.]]|volume=21|issue= 4|pages=1253–1278|doi= 10.1137/S0895479896305696|s2cid= 14344372|url= https://alterlab.org/teaching/BME6780/papers+patents/De_Lathauwer_2000.pdf}}</ref><ref name=Vasilescu2002Tensorfaces >{{cite book |first1=M.A.O. |last1=Vasilescu |first2=D. |last2=Terzopoulos |title=Computer Vision — ECCV 2002 |chapter=Multilinear Analysis of Image Ensembles: TensorFaces |series=Lecture Notes in Computer Science |volume=2350 |pages=447–460 |doi=10.1007/3-540-47969-4_30 |date=2002 |isbn=978-3-540-43745-1 |s2cid=12793247 |chapter-url=http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf |access-date=2022-12-29 |archive-date=2022-12-29 |archive-url=https://web.archive.org/web/20221229090931/http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf |url-status=dead }}</ref><ref name=KoldaBader2009 >{{cite journal| last1= Kolda |first1= Tamara| last2= Bader |first2= Brett| date=2009|title=Tensor Decompositions and Applications |journal= [[SIAM Review]]|volume=51|issue= 3|pages=455–500|doi= 10.1137/07070111X|bibcode= 2009SIAMR..51..455K|s2cid= 16074195|url= https://www.kolda.net/publication/TensorReview.pdf}}</ref> although the term "rank" generally has [[tensor rank|another meaning]] in the context of matrices and tensors.

Just as the components of a vector change when we change the [[basis (linear algebra)|basis]] of the vector space, the components of a tensor also change under such a transformation.  Each type of tensor comes equipped with a ''transformation law'' that details how the components of the tensor respond to a [[change of basis]].  The components of a vector can respond in two distinct ways to a [[change of basis]] (see ''[[Covariance and contravariance of vectors]]''), where the new [[basis vectors]] <math>\mathbf{\hat{e}}_i</math> are expressed in terms of the old basis vectors <math>\mathbf{e}_j</math> as,
:<math>\mathbf{\hat{e}}_i = \sum_{j=1}^n \mathbf{e}_j R^j_i = \mathbf{e}_j R^j_i .</math>

Here ''R''<sup>'' j''</sup><sub>''i''</sub> are the entries of the change of basis matrix, and in the rightmost expression the [[summation]] sign was suppressed: this is the [[Einstein summation convention]], which will be used throughout this article.<ref group="Note">The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term.  For example, under this convention <math>B_i C^i = B_1 C^1 + B_2 C^2 + \cdots + B_n C^n</math></ref>  The components ''v''<sup>''i''</sup> of a column vector '''v''' transform with the [[matrix inverse|inverse]] of the matrix ''R'',
:<math>\hat{v}^i = \left(R^{-1}\right)^i_j v^j,</math>

where the hat denotes the components in the new basis.  This is called a ''contravariant'' transformation law, because the vector components transform by the ''inverse'' of the change of basis.  In contrast, the components, ''w''<sub>''i''</sub>, of a covector (or row vector), '''w''', transform with the matrix ''R'' itself,
:<math>\hat{w}_i = w_j R^j_i .</math>

This is called a ''covariant'' transformation law, because the covector components transform by the ''same matrix'' as the change of basis matrix.  The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index.  If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called ''contravariant'' and is conventionally denoted with an upper index (superscript).  If the transformation matrix of an index is the basis transformation itself, then the index is called ''covariant'' and is denoted with a lower index (subscript).

As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array <math>T</math> that transforms under a change of basis matrix  <math>R = \left(R^j_i\right)</math> by <math>\hat{T} = R^{-1}TR</math>.  For the individual matrix entries, this transformation law has the form <math>\hat{T}^{i'}_{j'} = \left(R^{-1}\right)^{i'}_i T^i_j R^j_{j'}</math> so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).

Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above:

:<math>\mathbf{v} = \hat{v}^i \,\mathbf{\hat{e}}_i =
 \left( \left(R^{-1}\right)^i_j {v}^j \right) \left( \mathbf{{e}}_k R^k_i \right) =
 \left( \left(R^{-1}\right)^i_j R^k_i \right) {v}^j \mathbf{{e}}_k =
 \delta_j^k {v}^j  \mathbf{{e}}_k = {v}^k \,\mathbf{{e}}_k = {v}^i \,\mathbf{{e}}_i
</math>,

where <math>\delta^k_j</math> is the [[Kronecker delta]], which functions similarly to the [[identity matrix]], and has the effect of renaming indices (''j'' into ''k'' in this example). This shows several features of the component notation: the ability to re-arrange terms at will ([[commutativity]]), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like <math>{v}^i \,\mathbf{{e}}_i</math> can immediately be seen to be geometrically identical in all coordinate systems.

Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector.  The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations.  That is, the components <math>(Tv)^i</math> are given by  <math>(Tv)^i = T^i_j v^j</math>.  These components transform contravariantly, since
:<math>\left(\widehat{Tv}\right)^{i'} = \hat{T}^{i'}_{j'} \hat{v}^{j'} = \left[ \left(R^{-1}\right)^{i'}_i T^i_j R^j_{j'} \right] \left[ \left(R^{-1}\right)^{j'}_k v^k \right] = \left(R^{-1}\right)^{i'}_i (Tv)^i .</math>

The transformation law for an order {{math|''p'' + ''q''}} tensor with ''p'' contravariant indices and ''q'' covariant indices is thus given as,
:<math>
  \hat{T}^{i'_1, \ldots, i'_p}_{j'_1, \ldots, j'_q} = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
</math> <math>
  T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}
</math> <math>
  R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q}.
</math>

Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates.  Such a tensor is said to be of order or ''type'' {{math|(''p'', ''q'')}}.  The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept.  Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), {{math|''p'' + ''q''}} in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices.  A tensor of type {{math|(''p'', ''q'')}} is also called a {{math|(''p'', ''q'')}}-tensor for short.

This discussion motivates the following formal definition:<ref name="Sharpe2000">{{cite book|first=R.W. |last=Sharpe|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program|url={{google books |plainurl=y |id=Ytqs4xU5QKAC| page=194}}|date=2000|publisher=Springer |isbn=978-0-387-94732-7| page=194}}</ref><ref>{{citation|chapter-url={{google books |plainurl=y |id=WROiC9st58gC}}|first=Jan Arnoldus|last=Schouten|author-link=Jan Arnoldus Schouten|title=Tensor analysis for physicists|year=1954|publisher=Courier Corporation|isbn=978-0-486-65582-6|chapter=Chapter II|url=https://archive.org/details/isbn_9780486655826}}</ref>

{{blockquote|'''Definition.'''  A tensor of type (''p'', ''q'') is an assignment of a multidimensional array
:<math>T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}]</math>
to each basis {{math|'''f''' {{=}} ('''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>)}} of an ''n''-dimensional vector space such that, if we apply the change of basis
:<math>\mathbf{f}\mapsto \mathbf{f}\cdot R = \left( \mathbf{e}_i R^i_1, \dots, \mathbf{e}_i R^i_n \right)</math>
then the multidimensional array obeys the transformation law
:<math>
 T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
</math> <math>
  T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
</math> <math>
  R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} .
</math>
}}

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.<ref name="Kline" />

An equivalent definition of a tensor uses the [[representation theory|representations]] of the [[general linear group]].  There is an [[Group action (mathematics)|action]] of the general linear group on the set of all [[ordered basis|ordered bases]] of an ''n''-dimensional vector space.  If <math>\mathbf f = (\mathbf f_1, \dots, \mathbf f_n)</math> is an ordered basis, and <math>R = \left(R^i_j\right)</math> is an invertible <math>n\times n</math> matrix, then the action is given by
:<math>\mathbf fR = \left(\mathbf f_i R^i_1, \dots, \mathbf f_i R^i_n\right).</math>

Let ''F'' be the set of all ordered bases.  Then ''F'' is a [[principal homogeneous space]] for GL(''n'').  Let ''W'' be a vector space and let <math>\rho</math> be a representation of GL(''n'') on ''W'' (that is, a [[group homomorphism]] <math>\rho: \text{GL}(n) \to \text{GL}(W)</math>).  Then a tensor of type <math>\rho</math> is an [[equivariant map]] <math>T: F \to W</math>.  Equivariance here means that
:<math>T(FR) = \rho\left(R^{-1}\right)T(F).</math>

When <math>\rho</math> is a [[tensor representation]] of the general linear group, this gives the usual definition of tensors as multidimensional arrays.  This definition is often used to describe tensors on manifolds,<ref>{{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry|volume=1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=978-0-471-15733-5|title-link=Foundations of Differential Geometry}}</ref> and readily generalizes to other groups.<ref name="Sharpe2000" />

=== As multilinear maps ===
{{Main|Multilinear map}}
A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object.  Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred.  One approach that is common in [[differential geometry]] is to define tensors relative to a fixed (finite-dimensional) vector space ''V'', which is usually taken to be a particular vector space of some geometrical significance like the [[tangent space]] to a manifold.<ref>{{citation|last=Lee|first=John|title=Introduction to smooth manifolds|url={{google books |plainurl=y |id=4sGuQgAACAAJ|page=173}}|page=173|year=2000|publisher=Springer|isbn=978-0-387-95495-0}}</ref>  In this approach, a type {{nowrap|(''p'', ''q'')}} tensor ''T'' is defined as a [[multilinear map]],
:<math> T: \underbrace{V^* \times\dots\times V^*}_{p \text{ copies}} \times \underbrace{ V \times\dots\times V}_{q \text{ copies}} \rightarrow \mathbf{R}, </math>

where ''V''<sup>∗</sup> is the corresponding [[dual space]] of covectors, which is linear in each of its arguments. The above assumes ''V'' is a vector space over the [[real number]]s, {{tmath|\R}}. More generally, ''V'' can be taken over any [[Field (mathematics)|field]] ''F'' (e.g. the [[complex number]]s), with ''F'' replacing {{tmath|\R}} as the codomain of the multilinear maps.

By applying a multilinear map ''T'' of type {{nowrap|(''p'', ''q'')}} to a basis {'''e'''<sub>''j''</sub>} for ''V'' and a canonical cobasis {'''ε'''<sup>''i''</sup>} for ''V''<sup>∗</sup>,
:<math>T^{i_1\dots i_p}_{j_1\dots j_q} \equiv T\left(\boldsymbol{\varepsilon}^{i_1}, \ldots,\boldsymbol{\varepsilon}^{i_p}, \mathbf{e}_{j_1}, \ldots, \mathbf{e}_{j_q}\right),</math>

a {{nowrap|(''p'' + ''q'')}}-dimensional array of components can be obtained.  A different choice of basis will yield different components.  But, because ''T'' is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition.  The multidimensional array of components of ''T'' thus form a tensor according to that definition.  Moreover, such an array can be realized as the components of some multilinear map ''T''.  This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

In viewing a tensor as a multilinear map, it is conventional to identify the [[double dual]] ''V''<sup>∗∗</sup> of the vector space ''V'', i.e., the space of linear functionals on the dual  vector space ''V''<sup>∗</sup>, with the vector space ''V''.  There is always a [[Dual space#Injection into the double-dual|natural linear map]] from ''V'' to its double dual, given by evaluating a linear form in ''V''<sup>∗</sup> against a vector in ''V''.  This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify ''V'' with its double dual.

=== Using tensor products ===
{{Main|Tensor (intrinsic definition)}}
For some mathematical applications, a more abstract approach is sometimes useful.  This can be achieved by defining tensors in terms of elements of [[tensor product]]s of vector spaces, which in turn are defined through a [[universal property]] as explained [[Tensor product#Universal property|here]] and [[Tensor (intrinsic definition)#Universal property|here]].

A '''type {{math|(''p'', ''q'')}} tensor''' is defined in this context as an element of the tensor product of vector spaces,<ref>{{cite book|last1=Dodson|first1=C.T.J.|title=Tensor geometry: The Geometric Viewpoint and Its Uses |edition=2nd |date=2013 |isbn=9783642105142 |volume=130|orig-year=1991|series=Graduate Texts in Mathematics |publisher=Springer|last2=Poston |first2=T. |page= 105}}</ref><ref>{{Springer|id=a/a011120|title=Affine tensor}}</ref>
:<math>T \in \underbrace{V \otimes\dots\otimes V}_{p\text{ copies}} \otimes \underbrace{V^* \otimes\dots\otimes V^*}_{q \text{ copies}}.</math>

A basis {{math|''v''<sub>''i''</sub>}} of {{math|''V''}} and basis {{math|''w''<sub>''j''</sub>}} of {{math|''W''}} naturally induce a basis {{math|''v''<sub>''i''</sub> ⊗ ''w''<sub>''j''</sub>}} of the tensor product {{math|''V'' ⊗ ''W''}}.  The components of a tensor {{math|''T''}} are the coefficients of the tensor with respect to the basis obtained from a basis {{math|<nowiki>{</nowiki>'''e'''<sub>''i''</sub><nowiki>}</nowiki>}} for {{math|''V''}} and its dual basis {{math|{'''''ε'''''{{i sup|''j''}}<nowiki>}</nowiki>}}, i.e.
:<math>T = T^{i_1\dots i_p}_{j_1\dots j_q}\; \mathbf{e}_{i_1}\otimes\cdots\otimes \mathbf{e}_{i_p}\otimes \boldsymbol{\varepsilon}^{j_1}\otimes\cdots\otimes \boldsymbol{\varepsilon}^{j_q}.</math>

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type {{math|(''p'', ''q'')}} tensor.  Moreover, the universal property of the tensor product gives a [[bijection|one-to-one correspondence]] between tensors defined in this way and tensors defined as multilinear maps.

This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual:

:<math>U \otimes V \cong\left(U^{* *}\right) \otimes\left(V^{* *}\right) \cong\left(U^{*} \otimes V^{*}\right)^{*} \cong \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math>

The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from <math>\operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math> and <math>\operatorname{Hom}\left(U^{*} \otimes V^{*} ; \mathbb{F}\right)</math>.<ref>{{Cite web |date=June 5, 2021|title=Why are Tensors (Vectors of the form a⊗b...⊗z) multilinear maps? |url=https://math.stackexchange.com/q/4163471 |website=Mathematics Stackexchange}}</ref>

Tensor products can be defined in great generality&nbsp;– for example, [[tensor product of modules|involving arbitrary modules]] over a ring.  In principle, one could define a "tensor" simply to be an element of any tensor product.  However, the mathematics literature usually reserves the term ''tensor'' for an element of a tensor product of any number of copies of a single vector space {{math|''V''}} and its dual, as above.

=== Tensors in infinite dimensions ===
This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are [[naturally isomorphic]].<ref group="Note">The [[Dual space#Injection into the double-dual|double duality isomorphism]], for instance, is used to identify ''V'' with the double dual space ''V''<sup>∗∗</sup>, which consists of multilinear forms of degree one on ''V''<sup>∗</sup>.  It is typical in linear algebra to identify spaces that are naturally isomorphic, treating them as the same space.</ref>  Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to [[vector bundle]]s or [[coherent sheaves]].<ref>{{cite book|first=N. |last=Bourbaki|title=Algebra I: Chapters 1-3|chapter=3|chapter-url={{google books |plainurl=y |id=STS9aZ6F204C}}|date=1998|publisher=Springer |isbn=978-3-540-64243-5}} where the case of finitely generated projective modules is treated.  The global sections of sections of a vector bundle over a compact space form a projective module over the ring of smooth functions.  All statements for coherent sheaves are true locally.</ref>  For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see [[topological tensor product]]).  In some applications, it is the [[tensor product of Hilbert spaces]] that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a [[symmetric monoidal category]] that encodes their most important properties, rather than the specific models of those categories.<ref>{{citation|title= Braided tensor categories |first1= André |last1=Joyal |first2= Ross |last2=Street |journal= [[Advances in Mathematics]] |year=1993 |volume=102 |pages= 20–78 |doi= 10.1006/aima.1993.1055 |doi-access=free }}</ref>

=== Tensor fields ===
{{Main|Tensor field}}
In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space.  This was the setting of Ricci's original work.  In modern mathematical terminology such an object is called a [[tensor field]], often referred to simply as a tensor.<ref name="Kline" />

In this context, a [[coordinate basis]] is often chosen for the [[tangent space|tangent vector space]].  The transformation law may then be expressed in terms of [[partial derivative]]s of the coordinate functions,
:<math>\bar{x}^i\left(x^1, \ldots, x^n\right),</math>

defining a coordinate transformation,<ref name="Kline" />
:<math>
  \hat{T}^{i'_1\dots i'_p}_{j'_1\dots j'_q}\left(\bar{x}^1, \ldots, \bar{x}^n\right) =
  \frac{\partial \bar{x}^{i'_1}}{\partial x^{i_1}}
  \cdots
  \frac{\partial \bar{x}^{i'_p}}{\partial x^{i_p}}
  \frac{\partial x^{j_1}}{\partial \bar{x}^{j'_1}}
  \cdots
  \frac{\partial x^{j_q}}{\partial \bar{x}^{j'_q}}
  T^{i_1\dots i_p}_{j_1\dots j_q}\left(x^1, \ldots, x^n\right).
</math>