In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments. Thus a r^{th} order symmetric tensor represented in coordinates as a quantity with r indices satisfies
for every permutation σ of the symbols {1,2,...,r}.
The space of symmetric tensors of rank r on a finite dimensional vector space is naturally isomorphic to the dual of the space homogeneous polynomials of degree r on V. Over 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.
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Definition
Let V be a vector space and
a tensor of order r. Then T is a symmetric tensor if
for the braiding maps associated to every permutation σ on the symbols {1,2,...,r} (or equivalently for every transposition on these symbols).
Given a basis {e^{i}} of V, any symmetric tensor T of rank r can be written as
for some unique list of coefficients (the components of the tensor in the basis) that are symmetric on the indices. That is to say
for every permutation σ.
The space of all symmetric tensors of rank r defined on V is often denoted by S^{r}(V) or Sym^{r}(V). It is itself a vector space, and if V has dimension N then has Sym^{r}(V) has dimension
where is the binomial coefficient.
Symmetric part of a tensor
If is a tensor of order r, then the symmetric part of T is the symmetric tensor defined by
the summation extending over the symmetric group on r symbols. In terms of a basis, and employing the Einstein summation convention, if
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