Tensor: Difference between revisions

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=== 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''_''ij''}} , where {{mvar|i}} and {{mvar|j}} are indices running from {{math|1}} to {{mvar|n}}, or also by {{math|''T''  {{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''_''ij''}} and {{math|''T''  {{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 an array 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.

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:<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''^∗ 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, '''ℝ'''<math>\mathbb{R}</math>. More generally, ''V'' can be taken over any [[Field (mathematics)|field]] ''F'' (e.g. the [[complex number]]s), with ''F'' replacing '''ℝ'''<math>\mathbb{R}</math> as the codomain of the multilinear maps.

By applying a multilinear map ''T'' of type {{nowrap|(''p'', ''q'')}} to a basis {'''e'''_''j''} for ''V'' and a canonical cobasis {'''ε'''^''i''} for ''V''^∗,

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|issue=7–9

|issn=0302-7597

}} From p. 498: "And if we agree to call the ''square root'' (taken with a suitable sign) of this scalar product of two conjugate polynomes, P and KP, the common TENSOR of each, …... "</ref> to describe something different from what is now meant by a tensor.<ref group=Note>Namely, the [[norm (mathematics)|norm operation]] in a vector space.</ref> Gibbs introduced [[Dyadics]] and [[Polyadic algebra]], which are also tensors in the modern sense.<ref name="auto">{{Cite book |last=Guo |first=Hongyu |url=https://books.google.com/books?id=5dM3EAAAQBAJ&q=array+vector+matrix+tensor |title=What Are Tensors Exactly? |date=2021-06-16 |publisher=World Scientific |isbn=978-981-12-4103-1 |language=en}}</ref> The contemporary usage was introduced by [[Woldemar Voigt]] in 1898.<ref name="Voigt1898">{{cite book|first=Woldemar |last=Voigt|title=Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung |trans-title=The fundamental physical properties of crystals in an elementary presentation |url={{google books |plainurl=y |id=QhBDAAAAIAAJ|page=20}}|year=1898|publisher=Von Veit|pages=20–|quote= Wir wollen uns deshalb nur darauf stützen, dass Zustände der geschilderten Art bei Spannungen und Dehnungen nicht starrer Körper auftreten, und sie deshalb tensorielle, die für sie charakteristischen physikalischen Grössen aber Tensoren nennen. [We therefore want [our presentation] to be based only on [the assumption that] conditions of the type described occur during stresses and strains of non-rigid bodies, and therefore call them "tensorial" but call the characteristic physical quantities for them "tensors".]}}</ref>

Tensor calculus was developed around 1890 by [[Gregorio Ricci-Curbastro]] under the title ''absolute differential calculus'', and originally presented by Ricci-Curbastro in 1892.<ref>{{cite journal