WOLFRAM

Wolfram Language & System Documentation Center

TensorReduce[texpr]

attempts to return a canonical form for the symbolic tensor expression texpr.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Assumptions  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

TensorReduce

TensorReduce[texpr]

attempts to return a canonical form for the symbolic tensor expression texpr.

Details and Options

  • TensorReduce converts polynomials of symbolic tensor expressions containing arbitrary combinations of TensorProduct, TensorContract, and TensorTranspose into a canonical form with respect to symmetries.
  • If an expression is found to be equivalent to a zero tensor due to symmetry, the result will be 0.
  • If TensorDimensions[ten] does not return a list of dimensions, then the expression ten is returned unchanged.

Examples

open all close all

Basic Examples  (1)

Specify the properties of symbolic arrays:

Wolfram Language code: $Assumptions = {A∈Matrices[{3, 3}, Antisymmetric[{1, 2}]], S∈Arrays[{3, 3, 3}, Symmetric[{1, 2, 3}]]};

The trace of an antisymmetric matrix vanishes:

Wolfram Language code: TensorContract[A, {{1, 2}}]//TensorReduce

The contraction of a symmetric and an antisymmetric pair vanishes:

Wolfram Language code: TensorContract[AS, {{1, 3}, {2, 4}}]//TensorReduce

Reorder tensor products lexicographically:

Wolfram Language code: SA//TensorReduce

Scope  (2)

Canonicalization of symbolic tensor expressions:

Wolfram Language code: $Assumptions = {M∈Matrices[{3, 3}, Symmetric[{1, 2}]], (v | w)∈Vectors[{3}]};
Wolfram Language code: TensorContract[vMw + wMv, {{1, 2}, {3, 4}}]//TensorReduce

Mix explicit arrays and symbolic arrays:

Wolfram Language code: $Assumptions = A∈Matrices[{3, 3}, Symmetric[{1, 2}]];
Wolfram Language code: TensorTranspose[A]{1, 2, 3}//TensorReduce

Options  (1)

Assumptions  (1)

Specify local assumptions:

Wolfram Language code: TensorReduce[Transpose[A], Assumptions -> A∈Matrices[{3, 3}, Antisymmetric[{1, 2}]]]

Properties & Relations  (4)

Any transposition of a fully symmetric array is removed:

Wolfram Language code: $Assumptions = A∈Arrays[{d, d, d, d}, Symmetric[All]];
Wolfram Language code: TensorTranspose[A, RandomPermutation[4]]
Wolfram Language code: %//TensorReduce

For general symmetries, transpositions are converted into a canonical form:

Wolfram Language code: $Assumptions = A∈Arrays[{d, d, d, d}, {{{2, 1, 3, 4}, 1}, {{3, 4, 1, 2}, 1}}];
Wolfram Language code: TensorTranspose[A, #]& /@ Permutations[Range[4]]

In this case, there are three different canonical forms:

Wolfram Language code: TensorReduce /@ %//Union

A repeated tensor implies additional symmetry:

Wolfram Language code: $Assumptions = {A∈Arrays[{3, 3, 3}, Antisymmetric[{1, 2, 3}]], (v | w)∈Vectors[{3}]};
Wolfram Language code: TensorSymmetry[vwv]

Therefore, this contraction vanishes:

Wolfram Language code: TensorContract[Avwv, {{1, 4}, {2, 5}, {3, 6}}]//TensorReduce

For a given antisymmetric matrix, its contraction with itself n times gives 0 for odd n, but not for even n:

Wolfram Language code: $Assumptions = M∈Matrices[{d, d}, Antisymmetric[{1, 2}]];
Wolfram Language code: contractM[n_] := TensorContract[TensorProduct@@ConstantArray[M, n], Partition[RotateLeft[Range[2n]], 2]]
Wolfram Language code: contractM[1]
Wolfram Language code: contractM[2]
Wolfram Language code: contractM[3]
Wolfram Language code: Table[n -> TensorReduce[contractM[n]] === 0, {n, 20}]
Wolfram Research (2012), TensorReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorReduce.html.

Text

Wolfram Research (2012), TensorReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorReduce.html.

CMS

Wolfram Language. 2012. "TensorReduce." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TensorReduce.html.

APA

Wolfram Language. (2012). TensorReduce. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorReduce.html

BibTeX

@misc{reference.wolfram_2026_tensorreduce, author="Wolfram Research", title="{TensorReduce}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/TensorReduce.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_tensorreduce, organization={Wolfram Research}, title={TensorReduce}, year={2012}, url={https://reference.wolfram.com/language/ref/TensorReduce.html}, note=[Accessed: 12-June-2026]}