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Commutator — Definition, Formula & Examples

A commutator measures how much two elements fail to commute. If two elements commute (their order doesn't matter), the commutator equals the identity in a group or zero in a ring/algebra.

In group theory, the commutator of elements aa and bb in a group GG is defined as [a,b]=a−1b−1ab[a, b] = a^{-1}b^{-1}ab. In a ring or Lie algebra, the commutator of elements AA and BB is [A,B]=AB−BA[A, B] = AB - BA. The commutator equals the identity element (or zero, respectively) if and only if aa and bb commute.

Key Formula

[A,B]=AB−BA[A, B] = AB - BA
Where:
  • AA = First element (e.g., a matrix or ring element)
  • BB = Second element
  • [A,B][A, B] = The commutator, measuring the failure of A and B to commute

How It Works

To compute a commutator, you apply the appropriate formula depending on your algebraic structure. In group theory, you compose inverses and elements in the order a−1b−1aba^{-1}b^{-1}ab. In linear algebra or ring theory, you simply compute the difference AB−BAAB - BA. A zero or identity result tells you the elements commute; a nonzero result quantifies the failure of commutativity. The set of all commutators in a group generates the commutator subgroup, which captures the 'non-abelian part' of the group.

Worked Example

Problem: Compute the commutator [A, B] for the matrices A = [[1, 2], [0, 1]] and B = [[1, 0], [3, 1]].
Step 1: Compute the product AB.
AB=(1201)(1031)=(7231)AB = \begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1&0\\3&1\end{pmatrix} = \begin{pmatrix}7&2\\3&1\end{pmatrix}
Step 2: Compute the product BA.
BA=(1031)(1201)=(1237)BA = \begin{pmatrix}1&0\\3&1\end{pmatrix}\begin{pmatrix}1&2\\0&1\end{pmatrix} = \begin{pmatrix}1&2\\3&7\end{pmatrix}
Step 3: Subtract to find the commutator.
[A,B]=AB−BA=(600−6)[A,B] = AB - BA = \begin{pmatrix}6&0\\0&-6\end{pmatrix}
Answer: The commutator is [A,B]=(600−6)[A, B] = \begin{pmatrix}6&0\\0&-6\end{pmatrix}, which is nonzero, confirming that AA and BB do not commute.

Why It Matters

Commutators are central to quantum mechanics, where the commutator [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar encodes the Heisenberg uncertainty principle. In abstract algebra, the commutator subgroup determines whether a group is solvable, which connects directly to Galois theory and the unsolvability of the quintic equation.

Common Mistakes

Mistake: Confusing the group-theoretic commutator a−1b−1aba^{-1}b^{-1}ab with the ring/matrix commutator AB−BAAB - BA.
Correction: These are different formulas for different algebraic structures. Always check whether you are working in a group (use inverses and products) or a ring/algebra (use subtraction of products).