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

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Basic Examples  (1)

Specify the properties of symbolic arrays:

The trace of an antisymmetric matrix vanishes:

The contraction of a symmetric and an antisymmetric pair vanishes:

Reorder tensor products lexicographically:

Scope  (2)

Canonicalization of symbolic tensor expressions:

Mix explicit arrays and symbolic arrays:

Options  (1)

Assumptions  (1)

Specify local assumptions:

Properties & Relations  (4)

Any transposition of a fully symmetric array is removed:

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

In this case, there are three different canonical forms:

A repeated tensor implies additional symmetry:

Therefore, this contraction vanishes:

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

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.

BibTeX

@misc{reference.wolfram_2021_tensorreduce, author="Wolfram Research", title="{TensorReduce}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/TensorReduce.html}", note=[Accessed: 04-August-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_tensorreduce, organization={Wolfram Research}, title={TensorReduce}, year={2012}, url={https://reference.wolfram.com/language/ref/TensorReduce.html}, note=[Accessed: 04-August-2021 ]}

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