= Commutator
{title2=$[A,B]$}
{wiki}
= Matrix commutator
{synonym}
The commutator of two elements of an <associative algebra> over a <field> is
$$
[A,B]=AB-BA.
$$
It measures how much the order of multiplication matters: $[A,B]=0$ exactly when $AB=BA$. It is a <bilinear map>, it is alternating since $AA-AA=0$, and <associativity> makes it satisfy the <Jacobi identity>. Hence every <associative algebra> becomes a <Lie algebra> on the same <vector space> when equipped with this product. This holds in every <characteristic of a field>.
For square <matrices>, this is the matrix commutator, and it is defined even for singular <matrices>. The <group commutator> of invertible <matrices> is instead $ABA^{-1}B^{-1}$: it equals the <identity element> when the <matrices> commute, whereas $AB-BA$ equals the zero <matrix>. The <Lie bracket from local group commutators> explains how differentiation connects the two constructions.
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