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 just symmetric matrices, 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.

Contents
Definition
Examples
Properties
See also
References

Definition


A second-rank tensor is just a matrix. A matrix ''A'' , with components ''Aij'',
is said to be 'symmetric' if
:''Aij'' = ''Aji''
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 pi. 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.

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.

Properties


Any rank two tensor A_{ij}, can be represented as a sum of symmetric tensor and an antisymmetric tensor
: A_{ij} = rac{1}{2} (A_{ij}+A_{ji})+ rac{1}{2} (A_{ij}-A_{ji})
It is easily verified that the first term, denoted A_{(ij)}, does not change when indices are interchanged
: A_{(ij)} = rac{1}{2} (A_{ij}+A_{ji})= A_{(ji)}
While the second term, A_{[ij]},, picks up a minus sign.
: A_{[ij]} = rac{1}{2} (A_{ij}-A_{ji})= -A_{[ji]}
For a third order tensor, the symmetric & antisymmetric parts are
: A_{(ijk)} = rac{1}{3!} (A_{ijk}+A_{ikj}+A_{kij}+A_{kji}+A_{jki}+A_{jik})
: A_{[ijk]} = rac{1}{3!} (A_{ijk}-A_{ikj}+A_{kij}-A_{kji}+A_{jki}-A_{jik})
So for a general nth order tensor, the symmetric & antisymmetric parts are given by [1]
: T_{(mu_1 mu_2 mu_3 cdots mu _n)} = rac{1}{n!} sum_{j=0}^{n!-1} (mbox{permutations of } mu_1 cdots mu_n)
: T_{[mu_1 mu_2 mu_3 cdots mu _n]} = rac{1}{n!} sum_{j=0}^{n!-1} (-1)^j (mbox{permutations of } mu_1 cdots mu_n)
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
:mathrm{dim} , operatorname{Sym}^m(V)={n+m-1choose m}
where ''n'' is the dimension of ''V'' [2] and {n choose k} is the binomial coefficient.

See also



★ transpose

★ symmetric polynomial

★ Schur polynomial

★ Young symmetrizer

References


1. Sean M. Carroll, ''No-Nonsense Introduction to General Relativity (page 7)''
2. Cesar O. Aguilar, ''The Dimension of Symmetric k-tensors''


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