Symmetric tensor

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

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

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

T_{i_{1}i_{2}\cdots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma r}}.

The space of symmetric tensors of order 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},\ldots ,i_{k}=1}^{N}T_{i_{1}i_{2}\cdots i_{k}}e^{i_{1}}\otimes e^{i_{2}}\otimes \cdots \otimes e^{i_{k}}

for some unique list of coefficients $T_{i_{1}i_{2}\cdots i_{k}}$ (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}\cdots i_{\sigma k}}=T_{i_{1}i_{2}\cdots 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

There are many examples of symmetric tensors. Some include, the metric tensor, $g_{\mu \nu }$ , the Einstein tensor, $G_{\mu \nu }$ and the Ricci tensor, $R_{\mu \nu }$ .

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}\cdots 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}\cdots 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}\cdots i_{k})}={\frac {1}{k!}}\sum _{\sigma \in {\mathfrak {S}}_{k}}T_{i_{\sigma 1}i_{\sigma 2}\cdots 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_{{\sigma 1}}\otimes v_{{\sigma 2}}\otimes \cdots \otimes v_{{\sigma r}}.

In general we can turn Sym(V) into an algebra by defining the commutative and associative product ' $\odot$ '.^[1] 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^[1]) 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}}}}=\underbrace {v\otimes v\otimes \cdots \otimes v}_{{k{\text{ times}}}}=v^{{\otimes k}}.

Where v is a vector. Again, in some cases the ' $\odot$ ' is left out:

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

Decomposition

In analogy with the theory of symmetric matrices, a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor T ∈ Sym²(V), there are an integer r, non-zero unit vectors v₁,...,v_r ∈ V and weights λ₁,...,λ_r such that

T=\sum _{{i=1}}^{r}\lambda _{i}\,v_{i}\otimes v_{i}.

The minimum number r for which such a decomposition is possible is the (symmetric) rank of T. The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of inertia. Also see Sylvester's law of inertia.

For symmetric tensors of arbitrary order k, decompositions

T=\sum _{{i=1}}^{r}\lambda _{i}\,v_{i}^{{\otimes k}}

are also possible. The minimum number r for which such a decomposition is possible is the symmetric rank of T.^[2] This minimal decomposition is called a Waring decomposition; it is a symmetric form of the tensor rank decomposition. 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.

Notes

1 2 Kostrikin, Alexei I.; Manin, Iurii Ivanovich (1997). Linear algebra and geometry. Algebra, Logic and Applications. 1. Gordon and Breach. pp. 276–279. ISBN 9056990497.
↑ Comon, P.; Golub, G.; Lim, L. H.; Mourrain, B. (2008). "Symmetric Tensors and Symmetric Tensor Rank". SIAM Journal on Matrix Analysis and Applications. 30 (3): 1254. doi:10.1137/060661569.

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, MR 0224623 .
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

Tensors

Glossary of tensor theory

Scope

Mathematics	coordinate system multilinear algebra Euclidean geometry tensor algebra differential geometry exterior calculus tensor calculus

Physics Engineering	continuum mechanics electromagnetism transport phenomena general relativity computer vision

Notation

Tensor definitions

Operations

Related abstractions

Notable tensors

Mathematics	Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor

Physics	moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric tensor (GR)

Mathematicians

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