Symmetric Tensor

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