TensorSymmetry [tensor]
gives the symmetry of tensor under permutations of its slots.
TensorSymmetry [tensor,slots]
gives the symmetry under permutation of the specified list of slots.
TensorSymmetry
TensorSymmetry [tensor]
gives the symmetry of tensor under permutations of its slots.
TensorSymmetry [tensor,slots]
gives the symmetry under permutation of the specified list of slots.
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:
-
- The following options can be given:
-
- 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 all close allBasic Examples (2)
A symmetric matrix:
An antisymmetric matrix:
A symmetric array of rank 3:
Scope (7)
Find symmetry in an array:
Find symmetries in complex arrays:
Symmetry of a SymmetrizedArray object:
Symmetry of a SparseArray object:
Specify the symmetry of a symbolic array:
Symmetry of its tensor product with itself. Note the exchange symmetry:
A fully symmetric rank 3 array:
Complete symmetry:
Symmetry in a subset of slots:
Symmetry of an array of zeros:
Options (3)
Assumptions (1)
Specify locally the properties of the tensors:
With no assumptions, the symmetry is unknown:
SameTest (1)
This matrix is symmetric for a positive real , but TensorSymmetry gives no symmetry:
Use the option SameTest to get the correct answer:
Tolerance (1)
Generate a fully symmetric random array of depth 4:
The addition of a small perturbation breaks the symmetry:
The symmetry can be recovered by allowing some tolerance:
Properties & Relations (5)
Test whether a matrix is symmetric:
Find the symmetry of the matrix:
Test whether a matrix is antisymmetric:
Find the symmetry of the matrix:
Only a matrix of zeros can be simultaneously symmetric and antisymmetric:
Generation of special multidimensional symmetric arrays:
With a different radius, there are other symmetries:
The symmetry of Symmetrize [tensor,sym] is at least sym:
In some cases the result of Symmetrize [tensor,sym] may have more symmetry than sym:
Tech Notes
Related Guides
Text
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.
APA
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: 17-November-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: 17-November-2025]}