Operator conjugation by an exponential ($e^{S}H{e}^{-S}=H+[S,H]+\frac{1}{2}[S,[S,H]]+\cdots$)

For finite matrices, or formally on a common invariant domain where the expansion is meaningful, exponential conjugation is generated by the commutator map. Differentiating $F(s)=e^{sS}H{e}^{-sS}$ gives $F^{\prime }=[S,F]$, and its Taylor series yields the nested-commutator expansion. An anti-Hermitian $S$ makes this a unitary transformation. For perturbation theory this may be used order by order without asserting convergence for arbitrary unbounded operators.