= Operator conjugation by an exponential
{title2=$e^SHe^{-S}=H+[S,H]+\tfrac12[S,[S,H]]+\cdots$}
For finite <matrices>, or formally on a common invariant domain where the expansion is meaningful, exponential conjugation is generated by the <commutator> map. Differentiating $F(s)=e^{sS}He^{-sS}$ 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.
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