Symmetric tensor: Difference between revisions

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{{Short description|Tensor invariant under permutations of vectors it acts on}}

In [[mathematics]], a '''symmetric tensor''' is a [[tensor]] that is invariant under a [[permutation]] of its vector arguments:

{{Use American English|date = February 2019}}

:<math>T(v_1,v_2,\dots,v_r) = T(v_{\sigma 1},v_{\sigma 2},\dots,v_{\sigma r})</math>

In [[mathematics]], a '''symmetric tensor''' is aan [[Mixed tensor|unmixed]] [[tensor]] that is invariant under a [[permutation]] of its vector arguments:

for every permutation &sigma; of the symbols {1,2,...,''r''}.

Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies

:<math>T_{i_1i_2\dots i_r} = T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.</math>

:<math>T(v_1,v_2,\dotsldots,v_r) = T(v_{\sigma 1},v_{\sigma 2},\dotsldots,v_{\sigma r})</math>

The space of symmetric tensors of rank ''r'' on a finite-dimensional [[vector space]] is [[natural isomorphism|naturally isomorphic]] to the dual of the space of [[homogeneous polynomial]]s of degree ''r'' on ''V''. Over [[field (mathematics)|fields]] of [[characteristic zero]], the [[graded vector space]] of all symmetric tensors can be naturally identified with the [[symmetric algebra]] on ''V''. A related concept is that of the [[antisymmetric tensor]] or [[alternating form]]. Symmetric tensors occur widely in [[engineering]], [[physics]] and [[mathematics]].

for every permutation ''&sigma;'' of the symbols {{nowrap|{1, 2, ..., ''r''}.}} Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies

:<math>T_{i_1i_2\dotscdots i_r} = T_{i_{\sigma 1}i_{\sigma 2}\dotscdots i_{\sigma r}}.</math>

The space of symmetric tensors of rankorder ''r'' on a finite-dimensional [[vector space]] ''V'' is [[natural isomorphism|naturally isomorphic]] to the dual of the space of [[homogeneous polynomial]]s of degree ''r'' on ''V''. Over [[field (mathematics)|fields]] of [[characteristic zero]], the [[graded vector space]] of all symmetric tensors can be naturally identified with the [[symmetric algebra]] on ''V''. A related concept is that of the [[antisymmetric tensor]] or [[alternating form]]. Symmetric tensors occur widely in [[engineering]], [[physics]] and [[mathematics]].

==Definition==

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a tensor of order ''k''. Then ''T'' is a symmetric tensor if

:<math>\tau_\sigma T = T\,</math>

for the [[Tensor product#Tensor powers and braiding|braiding maps]] associated to every permutation ''&sigma;'' on the symbols {1,2,...,''k''} (or equivalently for every [[Transposition (mathematics)|transposition]] on these symbols).

Given a [[basis (linear algebra)|basis]] {''e''^''i''} of ''V'', any symmetric tensor ''T'' of rank ''k'' can be written as

:<math>T = \sum_{i_1,\dotsldots,i_k=1}^N T_{i_1i_2\dotscdots i_k} e^{i_1} \otimes e^{i_2}\otimes\cdots \otimes e^{i_k}</math>

for some unique list of coefficients <math>T_{i_1i_2\dotscdots i_k}</math> (the ''components'' of the tensor in the basis) that are symmetric on the indices. That is to say

:<math>T_{i_{\sigma 1}i_{\sigma 2}\dotscdots i_{\sigma k}} = T_{i_1i_2\dotscdots i_k}</math>

for every [[permutation]] ''&sigma;''.

The space of all symmetric tensors of order ''k'' defined on ''V'' is often denoted by ''S''^''k''(''V'') or Sym^''k''(''V''). It is itself a vector space, and if ''V'' has dimension ''N'' then the dimension of Sym^''k''(''V'') is the [[binomial coefficient]]

:<math>\dim\, \operatorname{Sym}^k(V) = {N + k - 1 \choose k}.</math>

We then construct Sym(''V'') as the [[direct sum of vector spaces|direct sum]] of Sym^''k''(''V'') for ''k'' = 0,1,2,…...

:<math>\operatorname{Sym}(V)= \bigoplus_{k=0}^\infty \operatorname{Sym}^k(V).</math>

==Examples==

There are many examples of symmetric tensors. Some include, the [[metric tensor]], <math>g_{\mu\nu}</math>, the [[Einstein tensor]], <math>G_{\mu\nu}</math> and the [[Ricci tensor]], <math>R_{\mu\nu}</math>.

Many [[material properties]] and [[field (physics)|fields]] used in physics and engineering can be represented as symmetric tensor fields; for example: [[stress (physics)|stress]], [[strain tensor|strain]], and [[anisotropic]] [[Electrical resistivity and conductivity|conductivity]]. Also, in [[diffusion MRI]] one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.

Ellipsoids are examples of [[algebraic varieties]]; and so, for general rank, symmetric tensors, in the guise of [[homogeneous polynomial]]s, are used to define [[projective varieties]], and are often studied as such.

Given a [[Riemannian manifold]] <math>(M,g)</math> equipped with its Levi-Civita connection <math>\nabla</math>, the [[Riemann curvature tensor#Coordinate expression|covariant curvature tensor]] is a symmetric order 2 tensor over the vector space <math display="inline">V = \Omega^2(M) = \bigwedge^2 T^*M</math> of differential 2-forms. This corresponds to the fact that, viewing <math>R_{ijk\ell} \in (T^*M)^{\otimes 4}</math>, we have the symmetry <math>R_{ij\, k\ell} = R_{k\ell\, ij}</math> between the first and second pairs of arguments in addition to antisymmetry within each pair: <math>R_{jik\ell} = - R_{ijk\ell} = R_{ij\ell k}</math>.<ref>{{Cite book |last=Carmo |first=Manfredo Perdigão do |title=Riemannian geometry |date=1992 |publisher=Birkhäuser |others=Francis J. Flaherty |isbn=0-8176-3490-8 |___location=Boston |oclc=24667701}}</ref>

==Symmetric part of a tensor==

Suppose <math>V</math> is a vector space over a field of [[Characteristic (algebra)|characteristic]] 0. If {{nowrap|''T'' &isin; ''V''^{&otimes;''k''}}} is a tensor of order <math>k</math>, then the symmetric part of <math>T</math> is the symmetric tensor defined by

:<math>\operatorname{Sym}\, T = \frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_k} \tau_\sigma T,</math>

the summation extending over the [[symmetric group]] on ''k'' symbols. In terms of a basis, and employing the [[Einstein summation convention]], if

:<math>T = T_{i_1i_2\dotscdots i_k}e^{i_1}\otimes e^{i_2}\otimes\cdots \otimes e^{i_k},</math>

then

:<math>\operatorname{Sym}\, T = \frac{1}{k!}\sum_{\sigma\in \mathfrak{S}_k} T_{i_{\sigma 1}i_{\sigma 2}\dotscdots i_{\sigma k}} e^{i_1}\otimes e^{i_2}\otimes\cdots \otimes e^{i_k}.</math>

The components of the tensor appearing on the right are often denoted by

:<math>T_{(i_1i_2\dotscdots i_k)} = \frac{1}{k!}\sum_{\sigma\in \mathfrak{S}_k} T_{i_{\sigma 1}i_{\sigma 2}\dotscdots i_{\sigma k}}</math>

with parentheses () around the indices which have beenbeing symmetrized. [Square brackets [] are used to indicate anti-symmetrization.]

==Symmetric product==

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:<math>T=v_1\otimes v_2\otimes\cdots \otimes v_r</math>

then the symmetric part of ''T'' is the symmetric product of the factors:

:<math>v_1\odot v_2\odot\cdots\odot v_r := \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} v_{i_{\sigma 1}}\otimes v_{i_{\sigma 2}}\otimes\cdots\otimes v_{i_{\sigma r}}.</math>

In general we can turn Sym(''V'') into an [[algebra]] by defining the commutative and associative product '<math>\odot</math>'⊙.<ref name="Kostrikin1997">{{cite book

| last1 = Kostrikin | first1 = Alexei I.

| last2 = Manin | first2 = Iurii Ivanovich

| authorlink1author-link1 = Alexei Kostrikin

| authorlink2author-link2 = Yuri I. Manin

| title = Linear algebra and geometry

| publisher = Gordon and Breach

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| pages = 276–279

| isbn = 9056990497

}}</ref> Given two tensors {{nowrap|''T''₁ &isin; Sym^''k''₁(''V'')}} and {{nowrap|''T''₂ &isin; Sym^''k''₂(''V'')}}, we use the symmetrization operator to define:

:<math>T_1\odot T_2 = \operatorname{Sym}(T_1\otimes T_2)\quad\left(\in\operatorname{Sym}^{k_1+k_2}(V)\right).</math>

It can be verified (as is done by Kostrikin and Manin<ref name="Kostrikin1997" />) that the resulting product is in fact commutative and associative. In some cases the operator is not written at allomitted: {{nowrap|1=''T''₁''T''₂ = ''T''₁<math>\odot</math> ⊙ ''T''₂}}.

In some cases an exponential notation is used:

:<math>v^{\odot k} = \underbrace{v \odot v \odot \cdots \odot v}_{k\text{ times}}=\underbrace{v \otimes v \otimes \cdots \otimes v}_{k\text{ times}}=v^{\otimes k}.</math>

Where ''v'' is a vector.

Again, in some cases the '<math>\odot</math>'⊙ is left out:

:<math>v^k=\underbrace{v\,v\,\cdots\,v}_{k\text{ times}}=\underbrace{v\odot v\odot\cdots\odot v}_{k\text{ times}}.</math>

==Decomposition==

In analogy with the theory of [[symmetric matrix|symmetric matrices]], a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor ''T''&nbsp;&isin;&nbsp;Sym²(''V''), there areis an integer ''r'', non-zero unit vectors ''v''₁,...,''v''_''r''&nbsp;&isin;&nbsp;''V'' and weights ''&lambda;''₁,...,''&lambda;''_''r'' such that

:<math>T = \sum_{i=1}^r \lambda_i \, v_i\otimes v_i.</math>

The minimum number ''r'' for which such a decomposition is possible is the (symmetric) rank of ''T''. The vectors appearing in this minimal expression are the ''[[Principal axis theorem|principal axes]]'' of the tensor, and generally have an important physical meaning. For example, the principal axes of the [[inertia tensor]] define the [[Poinsot's ellipsoid]] representing the moment of inertia. Also see [[Sylvester's law of inertia]].

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For symmetric tensors of arbitrary order ''k'', decompositions

:<math>T = \sum_{i=1}^r \lambda_i \, v_i^{\otimes k}</math>

are also possible. The minimum number ''r'' for which such a decomposition is possible is the ''symmetric'' [[Tensor_Tensor (intrinsic_definitionintrinsic definition)#Tensor_rankTensor rank|rank]] of ''T''.<ref name="Comon2008">{{Cite journal | last1 = Comon | first1 = P. | last2 = Golub | first2 = G. | last3 = Lim | first3 = L. H. | last4 = Mourrain | first4 = B. | title = Symmetric Tensors and Symmetric Tensor Rank | doi = 10.1137/060661569 | journal = SIAM Journal on Matrix Analysis and Applications | volume = 30 | issue = 3 | pages = 1254 | year = 2008 | arxiv = 0802.1681 | s2cid = 5676548 }}</ref> This minimal decomposition is called a Waring decomposition; it is a symmetric form of the [[tensor rank decomposition]]. For second -order tensors this corresponds to the rank of the matrix representing the tensor in any basis, and it is well- known that the maximum rank is equal to the dimension of the underlying vector space. However, for higher orders this need not hold: the rank can be higher than the number of dimensions in the underlying vector space. Moreover, Thethe [[higher-orderrank singularand valuesymmetric decomposition]]rank of a symmetric tensor ismay adiffer.<ref>{{Cite specialjournal|last=Shitov|first=Yaroslav|date=2018|title=A decompositionCounterexample ofto thisComon's formConjecture|url=https://epubs.siam.org/action/captchaChallenge?redirectUri=%2Fdoi%2F10.1137%2F17M1131970|journal=SIAM <refJournal name="Comon2008">{{Citeon doiApplied Algebra and Geometry|language=en-US|volume=2|issue=3|pages=428–443|doi=10.1137/06066156917m1131970|issn=2470-6566|arxiv=1705.08740|s2cid=119717133 }}</ref> (often called the [[CP decomposition|canonical decomposition]].)

==See also==

* [[antisymmetricAntisymmetric tensor]]

* [[Ricci calculus]]

* [[Schur polynomial]]

* [[symmetricSymmetric polynomial]]

* [[transposeTranspose]]

* [[Young symmetrizer]]

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==References==

* {{citation|first = Nicolas|last=Bourbaki|authorlinkauthor-link=Nicolas Bourbaki | title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989|isbn=3-540-64243-9}}.

* {{citation|first = Nicolas|last=Bourbaki|authorlinkauthor-link=Nicolas Bourbaki | title = Elements of mathematics, Algebra II| publisher = Springer-Verlag | year = 1990|isbn=3-540-19375-8}}.

* {{Citation | last1=Greub | first1=Werner Hildbert | title=Multilinear algebra | publisher=Springer-Verlag New York, Inc., New York | series=Die Grundlehren der Mathematischen Wissenschaften, Band 136 | idmr={{MathSciNet | id = 0224623}} | year=1967}}.

* {{Citation | last1=Sternberg | first1=Shlomo | author1-link=Shlomo Sternberg | title=Lectures on differential geometry | publisher=Chelsea | ___location=New York | isbn=978-0-8284-0316-0 | year=1983}}.

==External links==

* Cesar O. Aguilar, ''[https://web.archive.org/web/20061218155852/http://www.mast.queensu.ca/~cesar/math_notes/dim_symmetric_tensors.pdf The Dimension of Symmetric k-tensors]''

{{tensors}}