- Symmetric tensor
In
mathematics , a symmetric tensor is atensor that is invariant under apermutation of its vector arguments. Symmetric tensors of rank two are sometimes calledquadratic form s. In more abstract terms, symmetric tensors of general rank areisomorphic to the dual ofalgebraic form s; that is,homogeneous polynomial s and symmetric tensors are dual spaces. A related concept is that of theantisymmetric tensor oralternating form ; however, antisymmetric tensors have properties that are very different from those of symmetric tensors, and share little in common. Symmetric tensors occur widely inengineering ,physics andmathematics .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" withdual 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, andanisotropic conductivity . Symmetric rank 2 tensors can be diagonalized by choosing an orthogonal frame ofeigenvectors . These eigenvectors are the "principal axes " of the tensor, and generally have an important physical meaning. For example, the principal axes of themoment of inertia define theellipsoid representing the moment of inertia.Ellipsoids are examples of
algebraic varieties ; and so, for general rank, symmetric tensors, in the guise ofhomogeneous polynomial s, are used to defineprojective varieties , and are often studied as such.ee also
*
transpose
*symmetric polynomial
*Schur polynomial
*Young symmetrizer References
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