Symmetric tensor

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In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments. Thus a rth 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.

Contents

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 {ei} of V, any symmetric tensor T of rank r can be written as

for some unique list of coefficients T_{i_1i_2\dots i_r} (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 Sr(V) or Symr(V). It is itself a vector space, and if V has dimension N then has Symr(V) has dimension

where {a \choose b} is the binomial coefficient.

Symmetric part of a tensor

If \scriptstyle{T\in V^{\otimes r}} 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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