Symmetric tensor

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In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. Symmetric tensors of rank two are just symmetric matricies, and so are sometimes called quadratic forms. In more abstract terms, symmetric tensors of general rank are isomorphic to algebraic forms; that is, homogeneous polynomials and symmetric tensors are the same thing. A related concept is that of the antisymmetric tensor or alternating form; however, antisymmetric tensors have properties that are very different from those of symmetric tensors, and share little in common. Symmetric tensors occur widely in engineering, physics and mathematics.

1 Definition
2 Examples
3 Properties
4 See also
5 References

[edit] Definition

A second-rank tensor is just a matrix. A matrix A , with components A_ij, is said to be symmetric if

A_ij = A_ji

for all i, j. Using vector notation, a matrix is symmetric if, for vectors v and w, one has

A (v, w) = A (w, v)

Using tensor notation, given basis vectors $e i$ , their duals $e^*_i$ , one may write a matrix in terms of the tensor product of the dual basis as

$A=\sum_{i,j=1}^n A_{ij} e^*_i \otimes e^*_j$

and so, for a symmetric matrix, one has

$A(v \otimes w) = A(w \otimes v)$

More generally, the components of a symmetric tensor of rank m satisfy

$A_{i_1 i_2 \cdots i_m}= A_{i_{\pi(1)} i_{\pi(2)} \cdots i_{\pi (m)}}$

for any permutation $π$ . Equivalently, one may write

$A(v_1,v_2,\cdots,v_m) = A(v_{\pi(1)},v_{\pi(2)},\cdots,v_{\pi(m)})$

for vectors $v_1, v_2,\cdots$ .

[edit] 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. Symmetric rank 2 tensors can be diagonalized by choosing an orthogonal frame of eigenvectors. These eigenvectors are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the moment of inertia 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.

[edit] Properties

Any rank two tensor can be represented as a sum of symmetric tensor and antisymmetric tensor: $A = A s + A a$ , where

$A^s = \frac{1}{2} (A+A^T)$

and

$A^a = \frac{1}{2} (A - A^\mathrm{T})$

( $A T$ is the transpose of A: $A^T_{ij}=A_{ji}$ .)

The space of symmetric tensors of rank m defined on a vector space V is often denoted by $S m (V)$ or $\operatorname{Sym}^m(V)$ . This space has dimension