In mathematics, a symmetric tensor is an unmixed 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.
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,...
Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress, strain, and anisotropic conductivity.
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.
In terms of a basis, and employing the Einstein summation convention, if then The components of the tensor appearing on the right are often denoted by with parentheses () around the indices being symmetrized.
If T is a simple tensor, given as a pure tensor product then the symmetric part of T is the symmetric product of the factors: In general we can turn Sym(V) into an algebra by defining the commutative and associative product ⊙.
[2] Given two tensors T1 ∈ Symk1(V) and T2 ∈ Symk2(V), we use the symmetrization operator to define: It can be verified (as is done by Kostrikin and Manin[2]) that the resulting product is in fact commutative and associative.
Again, in some cases the ⊙ is left out: In analogy with the theory of symmetric matrices, a (real) symmetric tensor of order 2 can be "diagonalized".
More precisely, for any tensor T ∈ Sym2(V), there is an integer r, non-zero unit vectors v1,...,vr ∈ V and weights λ1,...,λr such that The minimum number r for which such a decomposition is possible is the (symmetric) rank of T. The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning.
For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of inertia.
For symmetric tensors of arbitrary order k, decompositions are also possible.
For second-order tensors this corresponds to the rank of the matrix representing the tensor in any basis, and it is well known that the maximum rank is equal to the dimension of the underlying vector space.
However, for higher orders this need not hold: the rank can be higher than the number of dimensions in the underlying vector space.