Symmetric tensor

In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. Symmetric tensors of rank two are sometimes called quadratic forms. In more abstract terms, symmetric tensors of general rank are isomorphic to the dual of algebraic forms; that is, homogeneous polynomials and symmetric tensors are dual spaces. 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.

Definition

Let "V" be a vector space of dimension "N" and "T" a tensor of rank "r" on "V". We call "T" a symmetric tensor if permuting its arguments does not change it, $forall\_\{s\; in\; mathrm\{Perm\}\_r\}\; T\; =\; T\; circ\; s$.

Given a basis $\{e\_i\}\_i$ of "V" with dual basis $\{e^i\}\_i$, a symmetric tensor "T" of rank 2 can in general be written as

:$T\; =\; sum\_\{i=1\}^N\; sum\_\{j=1\}^N\; T\_\{ij\}\; e^i\; otimes\; e^j$

and with Einstein summation convention that becomes

:$T\; =\; T\_\{ij\}\; e^i\; otimes\; e^j.$

The components "T"_"ij" of "T" form a symmetric matrix.

The space of all symmetric tensors of rank "r" defined on "V" is often denoted by $S^r(V)$ or $mathrm\{Sym\}^r(V)$ and has dimension

:$mathrm\{dim\}\; ,\; mathrm\{Sym\}^r(V)\; =\; \{N\; +\; r\; -\; 1\; choose\; r\}$ [Cesar O. Aguilar, " [http://www.mast.queensu.ca/~cesar/math_notes/dim_symmetric_tensors.pdf The Dimension of Symmetric k-tensors] "] where $\{a\; choose\; b\}$ is the binomial coefficient.

For convenience in writing symmetric tensors we define (the constant factor is sometimes chosen as 1)

:$a\; odot\; b\; :=\; (a\; otimes\; b\; +\; b\; otimes\; a)/2$

Homogeneous polynomials

The dual of $mathrm\{Sym\}^r(V)$ is isomorphic to the space of homogeneous polynomials of degree "r" on "V".

Let $f\; in\; mathrm\{Sym\}^2(V)$. Then $f\; =\; f^\{ij\}\; e\_i\; odot\; e\_j$ and its dual is $f^*\; =\; f\_\{ij\}\; e^i\; odot\; e^j$. The map $(mathrm\{Sym\}^r(V))^*\; =\; mathrm\{Sym\}^r(V^*)\; o\; mathrm\{Poly\}\_r(V)\; :\; f^*\; mapsto\; (v\; mapsto\; f^*(v,\; v,\; ldots,\; v))$ is an isomorphism of algebras.

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.

ee also

* transpose
* symmetric polynomial
* Schur polynomial
* Young symmetrizer

References