The _commutator_ is defined for a pair of [linear operators,](linear%20operator) $A$ and $B$ as $[A,B] = AB - BA.$ ^4d9559 This is a [skew-symmetric](Bilinear%20map.md#Symmetry%20properties%20of%20bilinear%20maps) [bilinear map.](Bilinear%20map.md) meaning that $[A,B]=-[B,A]$ ^89a345 Two operators, $A$ and $B,$ are said to _commute_ if $[A,B]=0$. ^b0c1a8 # Properties 1) $[A,B]^\dagger = [B^\dagger,A^\dagger]$ (where $^\dagger$ is the [adjoint](Adjoint.md)) 2) $(i[A,B])^\dagger = i[A,B]$, (i.e. $i[A,B]$ is a [Hermitian operator](Hermitian%20operators.md)) 3) $[AB,C] = A[B,C] + [A,C]B$ 4) $[A,BC] = [A,B]C + B[A,C]$ ^f3486a 5) $[A,[B,C]] = [B,[C,A]] = [C,[A,B]] = 0$ ([Jacobi identity](Jacobi%20identity.md)) ^f3fba1 6) $[A,F(A)] = 0$ where $F(A)$ is a matrix function of $A$. ^8c82a1 7) Define $F(B)$ as a polynomial in $B$ (i.e. a function of a linear operator written [as a power series](Functions%20of%20linear%20operators.md#as%20a%20power%20series)), where the derivative is $(B^n)=nB^{n-1}$. If $[A,[A,B]] = [B,[A,B]] = 0$ then $[A,F(B)] = [A,B]F'(B)$ ([proof](Commutators.md#Proof%20of%20property%207%20Commutators%20md%2081a4ed%20of%20commutators)). ^81a4ed # Commutators as Lie brackets The commutator is an example of an [adjoint map](Adjoint%20map.md) that satisfies the role of a bilinear map for many [Lie algebras](Lie%20algebras.md), hence why it's often referred to as a _[Lie bracket](Lie%20bracket.md)_ ([proof](Commutators.md#Proof%20that%20the%20commutator%20is%20an%20adjoint%20map%20of%20a%20Lie%20Algebra)). Two properties that allow [commutators](Commutators.md) to act as Lie brackets are [skew-symmetry](Commutators.md#^89a345) and [the Jacobi identity.](Commutators.md#^f3fba1)%%Isn't it sufficient to show that it is a Lie bracket by showing that it is antisymmetric and meets the jacobi identity? Also this header title needs to change at some point.%% # Commuting operators and matrices --- # Proofs and Examples ## Proof of [property 7](Commutators.md#^81a4ed) of commutators Since [$[A,[A,B]] = [B,[A,B]] = 0$](Commutators.md#^f3fba1) when can evaluate the outer brackets such that $A[A,B] = [A,B]A$ and $B[A,B] = [A,B]B$. Expanding $F(B)$ as a [as a power series](Functions%20of%20linear%20operators.md#as%20a%20power%20series): $F(B) = \sum_n^N F_n B^n \;\;\rightarrow \;\; F'(B) = \sum_n^N nF_n B^{n-1}$ which may be rewritten as $F(B) = F_0 + F_1 B + F_2 B^2 + ... F_nB^n$ Therefore: $[A,F(B)] = [A,\sum_n^N F_n B^n] = [A,F_0] + [A,F_1B] + [A,F_2B^2] + ...$ Next we show that $[A,B^n] = n[A,B]B^{n-1}$ Where we take the following base case: $[A,B^2]=[A,B]B + B[A,B] = 2[A,B]B$ since $A[A,B] = [A,B]A$ and $B[A,B] = [A,B]B$. Taking $n\rightarrow n+1$, $[A,B^{n+1}] = [A,BB^n] = -[BB^n,A] = -B[B^n,A] - [B,A]B^n = Bn[A,B]B^{n-1} - [B,A]B^n$ $=[A,B](nB^n+B^n)=[A,B]B^n(n+1)$ Therefore $[A,F(B)] = [A,B]F'(B)$. ## Proof that the commutator is an adjoint map of a Lie Algebra    #MathematicalFoundations/Algebra/AbstractAlgebra/Algebras #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices