The anti-commutator is a [bilinear map](Bilinear%20map.md) defined as $\{X,Y\} = XY + YX$ and it has the following properties: ^3bcda1 1) $\{A,B\}^\dagger = \{B^\dagger,A^\dagger\}$ (where $^\dagger$ is the [adjoint](Adjoint.md)) ^5448c1 2) $(i\{A,B\})^\dagger = -i\{A,B\}$ (i.e. $i\{A,B\}$ is an [anti-Hermitian operator](Anti-Hermitian%20operator.md)) ^b79676 #MathematicalFoundations/Algebra/AbstractAlgebra #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices