Symmetric tensor

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

$T(v_{1},v_{2},\dots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots ,v_{\sigma r})$

for every permutation σ of the symbols {1,2,...,r}. Alternatively, an r^th order symmetric tensor represented in coordinates as a quantity with r indices satisfies

$T_{i_{1}i_{2}\dots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.$

The space of symmetric tensors of rank r on a finite dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics.

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 braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols).

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

$T=\sum _{i_{1},\dots ,i_{k}=1}^{N}T_{i_{1}i_{2}\dots i_{k}}e^{i_{1}}\otimes e^{i_{2}}\otimes \cdots \otimes e^{i_{k}}$

for some unique list of coefficients $T_{i_{1}i_{2}\dots i_{k}}$ (the components of the tensor in the basis) that are symmetric on the indices. That is to say

$T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma k}}=T_{i_{1}i_{2}\dots i_{k}}$

for every permutation σ.

The space of all symmetric tensors of order k defined on V is often denoted by S^k(V) or Sym^k(V). It is itself a vector space, and if V has dimension N then the dimension of Sym^k(V) is the binomial coefficient

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

We then construct Sym(V) as the direct sum of Sym^k(V) for k = 0,1,2,…

$\operatorname {Sym} (V)=\bigoplus _{k=0}^{\infty }\operatorname {Sym} ^{k}(V).$

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. Also, in diffusion MRI one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.

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.

Symmetric part of a tensor

Suppose $V$ is a vector space over a field of characteristic 0. If T∈V^⊗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 symmetric group on k symbols. In terms of a basis, and employing the Einstein summation convention, if

$T=T_{i_{1}i_{2}\dots i_{k}}e^{i_{1}}\otimes e^{i_{2}}\otimes \cdots \otimes e^{i_{k}},$

then

$\operatorname {Sym} \,T={\frac {1}{k!}}\sum _{\sigma \in {\mathfrak {S}}_{k}}T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma k}}e^{i_{1}}\otimes e^{i_{2}}\otimes \cdots \otimes e^{i_{k}}.$

The components of the tensor appearing on the right are often denoted by

$T_{(i_{1}i_{2}\dots i_{k})}={\frac {1}{k!}}\sum _{\sigma \in {\mathfrak {S}}_{k}}T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma k}}$

with parentheses around the indices which have been 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:

$v_{1}\odot v_{2}\odot \cdots \odot v_{r}:={\frac {1}{r!}}\sum _{\sigma \in {\mathfrak {S}}_{r}}v_{i_{\sigma 1}}\otimes v_{i_{\sigma 2}}\otimes \cdots \otimes v_{i_{\sigma r}}.$

In general we can turn Sym(V) into an algebra by defining the commutative and associative product '$\odot$'. Given two tensors T₁∈Sym^k₁(V) and T₂∈Sym^k₂(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 not written at all: T₁T₂ = T₁$\odot$T₂.

In some cases an exponential notation is used:

$v^{\odot k}=\underbrace {v\odot v\odot \cdots \odot v} _{k{\text{ times}}}.$

Or simply:

$v^{k}=\underbrace {v\,v\,\cdots \,v} _{k{\text{ times}}}=\underbrace {v\odot v\odot \cdots \odot v} _{k{\text{ times}}}.$

Decomposition

For symmetric tensors of arbitrary order k, decompositions

$T=\sum _{i=1}^{r}\lambda _{i}\,\underbrace {v_{i}\otimes v_{i}\otimes \cdots \otimes v_{i}} _{k{\text{ times}}}$

are also possible. The minimum number r for which such a decomposition is possible is the symmetric rank of T. 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. The higher-order singular value decomposition of a symmetric tensor is a special decomposition of this form (often called the canonical decomposition.)

See also

• antisymmetric tensor
• Ricci calculus
• Schur polynomial
• symmetric polynomial
• transpose
• Young symmetrizer

References

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9.
• Bourbaki, Nicolas (1990), Elements of mathematics, Algebra II, Springer-Verlag, ISBN 3-540-19375-8.
• Greub, Werner Hildbert (1967), Multilinear algebra, Die Grundlehren der Mathematischen Wissenschaften, Band 136, Springer-Verlag New York, Inc., New York, MR0224623.
• Sternberg, Shlomo (1983), Lectures on differential geometry, New York: Chelsea, ISBN 978-0-8284-0316-0.

External links

• Cesar O. Aguilar, The Dimension of Symmetric k-tensors