TensorSymmetry
✖
TensorSymmetry
Details and Options

- TensorSymmetry accepts any type of tensor, either symbolic or explicit, including any type of array.
- A general symmetry is specified by a generating set of pairs {perm,ϕ}, where perm is a permutation of the slots of the tensor, and ϕ is a root of unity. Each pair represents a symmetry of the tensor of the form ϕ TensorTranspose[tensor,perm]==tensor.
- Some symmetry specifications have names:
-
Symmetric[{s1,…,sn}] full symmetry in the slots si Antisymmetric[{s1,…,sn}] antisymmetry in the slots si ZeroSymmetric[{s1,…,sn}] symmetry of a zero tensor - The following options can be given:
-
Assumptions $Assumptions assumptions to make about tensors SameTest Automatic function to test equality of expressions Tolerance Automatic tolerance for approximate numbers - For exact and symbolic arrays, the option SameTest->f indicates that two entries aij… and akl… are taken to be equal if f[aij…,akl…] gives True.
- For approximate arrays, the option Tolerance->t can be used to indicate that all entries Abs[aij…]≤t are taken to be zero.
- For array entries Abs[aij…]>t, equality comparison is done except for the last
bits, where
is $MachineEpsilon for MachinePrecision arrays and
for arrays of Precision
.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases

https://wolfram.com/xid/0rsy8amms-l1jrsp


https://wolfram.com/xid/0rsy8amms-le6d13


https://wolfram.com/xid/0rsy8amms-6qgilb

https://wolfram.com/xid/0rsy8amms-82w844

Scope (7)Survey of the scope of standard use cases

https://wolfram.com/xid/0rsy8amms-dbf8n0

Find symmetries in complex arrays:

https://wolfram.com/xid/0rsy8amms-u796hm

Symmetry of a SymmetrizedArray object:

https://wolfram.com/xid/0rsy8amms-wj2tue


https://wolfram.com/xid/0rsy8amms-2tswmd

Symmetry of a SparseArray object:

https://wolfram.com/xid/0rsy8amms-j2mgw5


https://wolfram.com/xid/0rsy8amms-ecq6v7

Specify the symmetry of a symbolic array:

https://wolfram.com/xid/0rsy8amms-z2l6p6

https://wolfram.com/xid/0rsy8amms-gnqc9d

Symmetry of its tensor product with itself. Note the exchange symmetry:

https://wolfram.com/xid/0rsy8amms-4mnhck

A fully symmetric rank 3 array:

https://wolfram.com/xid/0rsy8amms-glntma


https://wolfram.com/xid/0rsy8amms-f2c8kg

Symmetry in a subset of slots:

https://wolfram.com/xid/0rsy8amms-1z9c3k

Symmetry of an array of zeros:

https://wolfram.com/xid/0rsy8amms-fejnuo

Options (3)Common values & functionality for each option
Assumptions (1)
SameTest (1)
This matrix is symmetric for a positive real , but TensorSymmetry gives no symmetry:

https://wolfram.com/xid/0rsy8amms-em2aht

https://wolfram.com/xid/0rsy8amms-ej0bsf

Use the option SameTest to get the correct answer:

https://wolfram.com/xid/0rsy8amms-63zf1y

Tolerance (1)
Generate a fully symmetric random array of depth 4:

https://wolfram.com/xid/0rsy8amms-u97cw7

https://wolfram.com/xid/0rsy8amms-w74trk

https://wolfram.com/xid/0rsy8amms-zj1bl1

The addition of a small perturbation breaks the symmetry:

https://wolfram.com/xid/0rsy8amms-7ooegl

https://wolfram.com/xid/0rsy8amms-18r6kg

The symmetry can be recovered by allowing some tolerance:

https://wolfram.com/xid/0rsy8amms-e0wkny

Properties & Relations (5)Properties of the function, and connections to other functions
Test whether a matrix is symmetric:

https://wolfram.com/xid/0rsy8amms-zxssrt

https://wolfram.com/xid/0rsy8amms-f71jee

Find the symmetry of the matrix:

https://wolfram.com/xid/0rsy8amms-clwtjz

Test whether a matrix is antisymmetric:

https://wolfram.com/xid/0rsy8amms-j9fdyw

https://wolfram.com/xid/0rsy8amms-4zl6l

Find the symmetry of the matrix:

https://wolfram.com/xid/0rsy8amms-n6yc3x

Only a matrix of zeros can be simultaneously symmetric and antisymmetric:

https://wolfram.com/xid/0rsy8amms-yueai8


https://wolfram.com/xid/0rsy8amms-ssw1g9


https://wolfram.com/xid/0rsy8amms-uwf0n4


https://wolfram.com/xid/0rsy8amms-fr8txh

Generation of special multidimensional symmetric arrays:

https://wolfram.com/xid/0rsy8amms-kgrwe2


https://wolfram.com/xid/0rsy8amms-b3k6pu


https://wolfram.com/xid/0rsy8amms-k3ejp4

With a different radius, there are other symmetries:

https://wolfram.com/xid/0rsy8amms-idgrpv


https://wolfram.com/xid/0rsy8amms-0in36m

The symmetry of Symmetrize[tensor,sym] is at least sym:

https://wolfram.com/xid/0rsy8amms-mi1qre

https://wolfram.com/xid/0rsy8amms-6zl05b


https://wolfram.com/xid/0rsy8amms-fyhy2p

In some cases the result of Symmetrize[tensor,sym] may have more symmetry than sym:

https://wolfram.com/xid/0rsy8amms-z4l4ss


https://wolfram.com/xid/0rsy8amms-fav4xu

Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).
Text
Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).
Wolfram Research (2012), TensorSymmetry, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorSymmetry.html (updated 2017).
CMS
Wolfram Language. 2012. "TensorSymmetry." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/TensorSymmetry.html.
Wolfram Language. 2012. "TensorSymmetry." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/TensorSymmetry.html.
APA
Wolfram Language. (2012). TensorSymmetry. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorSymmetry.html
Wolfram Language. (2012). TensorSymmetry. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorSymmetry.html
BibTeX
@misc{reference.wolfram_2025_tensorsymmetry, author="Wolfram Research", title="{TensorSymmetry}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/TensorSymmetry.html}", note=[Accessed: 04-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_tensorsymmetry, organization={Wolfram Research}, title={TensorSymmetry}, year={2017}, url={https://reference.wolfram.com/language/ref/TensorSymmetry.html}, note=[Accessed: 04-June-2025
]}