Symmetric tensor
Tensor invariant under permutations of vectors it acts on / From Wikipedia, the free encyclopedia
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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.