Symmetric tensor

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

Contents 1 Definition 1.1 Symmetric part of a tensor 2 Examples 3 See also 4 Notes 5 References 6 External links

Definition

Let V be a vector space and

a tensor of order r. Then T is a symmetric tensor if

τ_σT = T

for the braiding maps associated to every permutation σ on the symbols {1,2,...,r} (or equivalently for every transposition on these symbols).

Given a basis {eⁱ} of V, any symmetric tensor T of rank r 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 rank r defined on V is often denoted by S^r(V)</math> or Sym^r(V). It is itself a vector space, and if V has dimension N then has Symⁿ(V) has dimension

where is the binomial coefficient.

The space of symmetric tensors of degree r can equivalently be characterized as the kernel of the quotient map

from the r^th tensor power of V to the r^th exterior power of V.^[1]

Symmetric part of a tensor

If is a tensor of order r, then the symmetric part of T is the symmetric tensor defined by

the summation extending over the symmetric group on r symbols. 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

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:

Examples

Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example, stress, strain, and anisotropic conductivity.

In full 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 n and non-zero vectors v₁,...,v_n ∈ V such that

This is Sylvester's law of inertia. The minimum number n for which such a decomposition is possible is the 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 ellipsoid representing the moment of inertia.

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.

See also

• transpose
• symmetric polynomial
• Schur polynomial
• Young symmetrizer

Notes

1. ^ Greub 1967, Chapter VI, §3

References

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 .
• Greub, Werner Hildbert (1967), Multilinear algebra, Die Grundlehren der Mathematischen Wissenschaften, Band 136, Springer-Verlag New York, Inc., New York, MR0224623 .
• Sternberg, Shlomo (1983), Lectures on differential geometry, New York: Chelsea, ISBN 978-0-8284-0316-0 .

External links

• Cesar O. Aguilar, The Dimension of Symmetric k-tensors