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Encyclopedia > 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 matricies, 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In the mathematics of the nineteenth century, an important role was played by the algebraic forms that generalise quadratic forms to degrees 3 and more, also known as quantics. ... In mathematics, homogeneous has a variety of meanings. ... In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = &#8722;A or in component form, if A = (aij): aij = &#8722; aji   for all i and j. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the branch of science concerned with the fundamental laws of the universe. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

## Contents

A second-rank tensor is just a matrix. A matrix A , with components Aij, is said to be symmetric if Rank means a wide variety of things in mathematics, including: Rank (linear algebra) Rank of a tensor Rank of an array Rank of an abelian group Rank (set theory) Rank-into-rank Rank of a greedoid This is a disambiguation page &#8212; a navigational aid which lists other pages that... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

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 ei, their duals $e^*_i$, one may write a matrix in terms of the tensor product of the dual basis as In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In linear algebra, a dual basis is a set of orthogonal vectors that span (i. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

$A=sum_{i,j=1}^n A_{ij} e^*_i otimes e^*_j$

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 π. Equivalently, one may write Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...

$A(v_1,v_2,cdots,v_m) = A(v_{pi(1)},v_{pi(2)},cdots,v_{pi(m)})$

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. This is a list of materials properties. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... The strain tensor, Îµ, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients Îµii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms Îµij = 1/2 Î³ij (i... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... Electrical conductivity is a measure of how well a material accommodates the transport of electric charge. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ...

## Properties

Any rank two tensor can be represented as a sum of symmetric tensor and antisymmetric tensor: A = As + Aa, where In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...

$A^s = frac{1}{2} (A+A^T)$

and

$A^a = frac{1}{2} (A - A^mathrm{T})$

(AT is the transpose of A: $A^T_{ij}=A_{ji}$.) In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...

The space of symmetric tensors of rank m defined on a vector space V is often denoted by Sm(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 [1] and ${n choose k}$ is the binomial coefficient. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ... In commutative algebra and invariant theory, Schur polynomials are certain homogeneous symmetric polynomials. ... In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that the image of the element corresponds to an irreducible representation of the symmetric group. ...

## References

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

Results from FactBites:

 symmetric tensor: Definition and Much More from Answers.com (402 words) 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. 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.
 Springer Online Reference Works (287 words) A tensor which does not change on symmetrization with respect to some group of indices is called a symmetric tensor. Symmetrization, with respect to some group, of a tensor which was alternated first (see Alternation) with respect to that group, leads to the zero tensor. Symmetrization of tensors, side by side with the alternation operation, is used for the decomposition of a tensor into tensors with a simpler structure.
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