Let , , ... be operators. Then the commutator of and is defined as

(1)

Let , , ... be constants. Identities include

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Let and be tensors. Then

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There is a related notion of commutator in the theory of groups. The commutator of two group elements and is , and two elements and are said to commute when their commutator is the identity element. When the group is a Lie group, the Lie bracket in its Lie algebra is an infinitesimal version of the group commutator. For instance, let and be square matrices, and let and be paths in the Lie group of nonsingular matrices which satisfy

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then

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Portions of this entry contributed by Todd Rowland

CITE THIS AS:

Rowland, Todd and Weisstein, Eric W. "Commutator." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Commutator.html