Symmetric tensor

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

for every permutation σ of the symbols {1,2,...,r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies

The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of 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.

Let V be a vector space and

a tensor of order k. Then T is a symmetric tensor if

for the braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols).

Given a basis {eⁱ} of V, any symmetric tensor T of rank k can be written as

for some unique list of coefficients ${\displaystyle T_{i_{1}i_{2}\cdots i_{k}}}$ (the components of the tensor in the basis) that are symmetric on the indices. That is to say

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