Commutation

Commutation is a concept used in mathematics and physics to describe whether the order in which operations are performed affects the result. In algebra, a binary operation is commutative if a product a*b equals b*a for all elements a and b. When this holds, the order of combining elements is irrelevant.

Common examples include real-number addition and multiplication, both of which are commutative. By contrast, matrix multiplication is generally noncommutative; matrices A and B may satisfy AB ≠ BA.

To quantify non-commutativity, one uses the commutator [A,B] = AB − BA. If [A,B] = 0, the two elements A and B commute. In group theory, the commutator of two elements g and h is defined as g^{-1}h^{-1}gh, and the subgroup generated by all such commutators is the commutator (or derived) subgroup. This subgroup measures how far the group is from being abelian.

In Lie algebras, the Lie bracket [X,Y] serves a related role, encoding the noncommutativity of infinitesimal transformations and obeying bilinearity, antisymmetry, and the Jacobi identity.

In quantum mechanics and other areas of physics, observables are represented by operators. When two operators do not commute, their measurement outcomes depend on order, and uncertainty relations arise; a famous example is the canonical commutation relation [x,p] = iħ, underpinning the Heisenberg principle.

If commuting operators are also diagonalizable, they can often be simultaneously diagonalized, simplifying analysis. The set of all elements that commute with a given element—its centralizer—plays a key role in understanding algebraic structure.