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Concept
Symmetric tensor
Summary
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.
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 (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 order k defined on V is often denoted by Sk(V) or Symk(V). It is itself a vector space, and if V has dimension N then the dimension of Symk(V) is the binomial coefficient
We then construct Sym(V) as the direct sum of Symk(V) for k = 0,1,2,...
There are many examples of symmetric tensors. Some include, the metric tensor, , the Einstein tensor, and the Ricci tensor, .
Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress, strain, and anisotropic conductivity. Also, in diffusion MRI one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.
Ellipsoids are examples of algebraic varieties; and so, for general rank, symmetric tensors, in the guise of homogeneous polynomials, are used to define projective varieties, and are often studied as such.
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