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 order ''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 order ''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>

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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>v_1\odot v_2\odot\cdots\odot v_r := \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} v_{\sigma 1}\otimes v_{\sigma 2}\otimes\cdots\otimes v_{\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 (intrinsic definition)#Tensor 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, the rank and symmetric rank of a symmetric tensor may differ.<ref>{{Cite journal|last=Shitov|first=Yaroslav|date=2018|title=A Counterexample to Comon's Conjecture|url=https://epubs.siam.org/action/captchaChallenge?redirectUri=%2Fdoi%2F10.1137%2F17M1131970|journal=SIAM Journal on Applied Algebra and Geometry|language=en-US|volume=2|issue=3|pages=428–443|doi=10.1137/17m1131970|issn=2470-6566|arxiv=1705.08740|s2cid=119717133 }}</ref>

==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 |mr=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}}