BUILT-IN MATHEMATICA SYMBOL

NonCommutativeMultiply

is a general associative, but non-commutative, form of multiplication.

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.

Compare commutative multiplication with non-commutative multiplication:

Operations are associative:

Compare commutative multiplication with non-commutative multiplication:

Out[1]=

Operations are associative:

Out[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:

No automatic simplification rules exist for NonCommutativeMultiply:

Expand and Simplify do not operate on expressions with NonCommutativeMultiply:

NonCommutativeMultiply of one argument, unlike Times, stays unevaluated: