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Commutator[x,y]

gives the commutator x**y-y**x of x and y.

Commutator[x,y,alg]

gives the commutator of x and y in the noncommutative algebra alg.

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Commutator

Commutator[x,y]

gives the commutator x**y-y**x of x and y.

Commutator[x,y,alg]

gives the commutator of x and y in the noncommutative algebra alg.

Details

Examples

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Basic Examples  (2)

The commutator of x and y over an algebra with the default operations:

Wolfram Language code: Commutator[x, y]

The commutator of x and y over an algebra with symbolic operations:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Multiplication" -> mult, "Addition" -> add|>]; Commutator[x, y, alg]

Scope  (4)

The commutator of x and y over the algebra of square matrices with Dot product:

Wolfram Language code: Commutator[x, y, {Dot, n}]

The commutator of x and y over the algebra of linear endomorphisms with Composition:

Wolfram Language code: Commutator[x, y, Composition]

Traditional form of Commutator[x,y]:

Wolfram Language code: Hold[Commutator[x, y]]//TraditionalForm

Use the traditional form as input:

Wolfram Language code: Commutator[x, y]

Reducing a polynomial modulo the commutator of two variables commutes the variables:

Wolfram Language code: NonCommutativePolynomialReduce[x**y**x**y**x + 2y**y**x**x**y**y, {Commutator[x, y]}, {y, x}][[2]]

Properties & Relations  (2)

Reducing a polynomial modulo the commutator of two variables commutes the variables:

Wolfram Language code: NonCommutativePolynomialReduce[2x**y**x + 3x**x**y**x**x**y**y, {Commutator[x, y]}, {y, x}][[2]]

Anticommutator gives the anticommutator of two elements of an algebra:

Wolfram Language code: Anticommutator[x, y]
Wolfram Research (2025), Commutator, Wolfram Language function, https://reference.wolfram.com/language/ref/Commutator.html.

Text

Wolfram Research (2025), Commutator, Wolfram Language function, https://reference.wolfram.com/language/ref/Commutator.html.

CMS

Wolfram Language. 2025. "Commutator." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Commutator.html.

APA

Wolfram Language. (2025). Commutator. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Commutator.html

BibTeX

@misc{reference.wolfram_2026_commutator, author="Wolfram Research", title="{Commutator}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/Commutator.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_commutator, organization={Wolfram Research}, title={Commutator}, year={2025}, url={https://reference.wolfram.com/language/ref/Commutator.html}, note=[Accessed: 12-June-2026]}