What is 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 V 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.