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{{Short description|Tensor invariant under permutations of vectors it acts on}}
{{Use American English|date = February 2019}}
In [[mathematics]], a '''symmetric tensor''' is
:<math>T(v_1,v_2,\ldots,v_r) = T(v_{\sigma 1},v_{\sigma 2},\ldots,v_{\sigma r})</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==
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==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'' ∈ Sym<sup>2</sup>(''V''), there
:<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==
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