# 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: