is a general associative, but noncommutative, form of multiplication.


  • NonCommutativeMultiply has attribute Flat.
  • Instances of NonCommutativeMultiply are automatically flattened, but no other simplification is performed.
  • You can use NonCommutativeMultiply as a generalization of ordinary multiplication for special mathematical objects.


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Basic Examples  (1)

Compare commutative multiplication with non-commutative multiplication:

Operations are associative:

Applications  (2)

Use NonCommutativeMultiply to represent composition in an algebra of differential operators.

The base case, where is a function, simply multiplies by :

The next two properties express linearity:

Here the operator is D. HoldPattern stops the derivative from acting on the double blank:

Composition of operators applied to an expression:

Power of an operator applied to an expression:

Apply these rules to derive the KdV equation for the Lax pair:

Build a function to expand non-commutative products. Distributivity with respect to Plus:

Handling the commutative product inside the non-commutative one:

Fall-back operation applied to everything else:

Properties & Relations  (2)

No automatic simplification rules exist for NonCommutativeMultiply:

Expand and Simplify do not operate on expressions with NonCommutativeMultiply:

Possible Issues  (1)

NonCommutativeMultiply of one argument, unlike Times, stays unevaluated:

Wolfram Research (1988), NonCommutativeMultiply, Wolfram Language function,


Wolfram Research (1988), NonCommutativeMultiply, Wolfram Language function,


Wolfram Language. 1988. "NonCommutativeMultiply." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). NonCommutativeMultiply. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_noncommutativemultiply, author="Wolfram Research", title="{NonCommutativeMultiply}", year="1988", howpublished="\url{}", note=[Accessed: 21-April-2024 ]}


@online{reference.wolfram_2023_noncommutativemultiply, organization={Wolfram Research}, title={NonCommutativeMultiply}, year={1988}, url={}, note=[Accessed: 21-April-2024 ]}