Commutator

$[A,B]:=AB-BA$ is defined to be the commutator between Operators $A$ and $B$.

Properties

• $[A,B]=-[B,A]$: Commutators are anti-commutative
• $[xA+yB,C]=x[A,C]+y[B,C]$: Commutators are bilinear
• $[A,BC]=[A,B]C+B[A,C]$: Matrix multiplication distributes over commutation
• $[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$: Commutators satisfy the Jacobi Identity