Symmetric tensor

In, a symmetric tensor is a that is invariant under a  of its vector arguments:


 * $$T(v_1,v_2,\ldots,v_r) = T(v_{\sigma 1},v_{\sigma 2},\ldots,v_{\sigma r})$$

for every permutation &sigma; of the symbols {1, 2, ..., r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies
 * $$T_{i_1i_2\cdots i_r} = T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma r}}.$$

The space of symmetric tensors of order r on a finite-dimensional is  to the dual of the space of s of degree r on V. Over of, the  of all symmetric tensors can be naturally identified with the  on V. A related concept is that of the or. Symmetric tensors occur widely in, and.

Definition
Let V be a vector space and
 * $$T\in V^{\otimes k}$$

a tensor of order k. Then T is a symmetric tensor if
 * $$\tau_\sigma T = T\,$$

for the associated to every permutation &sigma; on the symbols {1,2,...,k} (or equivalently for every  on these symbols).

Given a {ei} of V, any symmetric tensor T of rank k can be written as


 * $$T = \sum_{i_1,\ldots,i_k=1}^N T_{i_1i_2\cdots i_k} e^{i_1} \otimes e^{i_2}\otimes\cdots \otimes e^{i_k}$$

for some unique list of coefficients $$T_{i_1i_2\cdots i_k}$$ (the components of the tensor in the basis) that are symmetric on the indices. That is to say


 * $$T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma k}} = T_{i_1i_2\cdots i_k}$$

for every &sigma;.

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


 * $$\dim\operatorname{Sym}^k(V) = {N + k - 1 \choose k}.$$

We then construct Sym(V) as the of Symk(V) for k = 0,1,2,...
 * $$\operatorname{Sym}(V)= \bigoplus_{k=0}^\infty \operatorname{Sym}^k(V).$$

Examples
There are many examples of symmetric tensors. Some include, the, $$g_{\mu\nu}$$, the , $$G_{\mu\nu}$$ and the , $$R_{\mu\nu}$$.

Many and  used in physics and engineering can be represented as symmetric tensor fields; for example:, , and. Also, in one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.

Ellipsoids are examples of ; and so, for general rank, symmetric tensors, in the guise of s, are used to define, and are often studied as such.

Symmetric part of a tensor
Suppose $$V$$ is a vector space over a field of 0. If T &isin; V&otimes;k is a tensor of order $$k$$, then the symmetric part of $$T$$ is the symmetric tensor defined by
 * $$\operatorname{Sym}\, T = \frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_k} \tau_\sigma T,$$

the summation extending over the on k symbols. In terms of a basis, and employing the, if
 * $$T = T_{i_1i_2\cdots i_k}e^{i_1}\otimes e^{i_2}\otimes\cdots \otimes e^{i_k},$$

then
 * $$\operatorname{Sym}\, T = \frac{1}{k!}\sum_{\sigma\in \mathfrak{S}_k} T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma k}} e^{i_1}\otimes e^{i_2}\otimes\cdots \otimes e^{i_k}.$$

The components of the tensor appearing on the right are often denoted by
 * $$T_{(i_1i_2\cdots i_k)} = \frac{1}{k!}\sum_{\sigma\in \mathfrak{S}_k} T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma k}}$$

with parentheses around the indices being symmetrized. Square brackets [] are used to indicate anti-symmetrization.

Symmetric product
If T is a simple tensor, given as a pure tensor product
 * $$T=v_1\otimes v_2\otimes\cdots \otimes v_r$$

then the symmetric part of T is the symmetric product of the factors:
 * $$v_1\odot v_2\odot\cdots\odot v_r := \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} v_{\sigma 1}\otimes v_{\sigma 2}\otimes\cdots\otimes v_{\sigma r}.$$

In general we can turn Sym(V) into an by defining the commutative and associative product ⊙. Given two tensors T1 &isin; Symk1(V) and T2 &isin; Symk2(V), we use the symmetrization operator to define:
 * $$T_1\odot T_2 = \operatorname{Sym}(T_1\otimes T_2)\quad\left(\in\operatorname{Sym}^{k_1+k_2}(V)\right).$$

It can be verified (as is done by Kostrikin and Manin) that the resulting product is in fact commutative and associative. In some cases the operator is omitted: T1T2 = T1 ⊙ T2.

In some cases an exponential notation is used:
 * $$v^{\odot k} = \underbrace{v \odot v \odot \cdots \odot v}_{k\text{ times}}=\underbrace{v \otimes v \otimes \cdots \otimes v}_{k\text{ times}}=v^{\otimes k}.$$

Where v is a vector. Again, in some cases the ⊙ is left out:
 * $$v^k=\underbrace{v\,v\,\cdots\,v}_{k\text{ times}}=\underbrace{v\odot v\odot\cdots\odot v}_{k\text{ times}}.$$

Decomposition
In analogy with the theory of, a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor T &isin; Sym2(V), there are an integer r, non-zero unit vectors v1,...,vr &isin; V and weights &lambda;1,...,&lambda;r such that
 * $$T = \sum_{i=1}^r \lambda_i \, v_i\otimes v_i.$$

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  of the tensor, and generally have an important physical meaning. For example, the principal axes of the define the  representing the moment of inertia. Also see.

For symmetric tensors of arbitrary order k, decompositions
 * $$T = \sum_{i=1}^r \lambda_i \, v_i^{\otimes k}$$

are also possible. The minimum number r for which such a decomposition is possible is the symmetric of T. This minimal decomposition is called a Waring decomposition; it is a symmetric form of the. 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. Moreover, the rank and symmetric rank of a symmetric tensor may differ.