Symmetric tensor: Difference between revisions

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

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

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\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]] 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]].