= Symmetrized exponential-splitting defect identity
{title2=$F(t)=\tfrac12(e^{tA}e^{tB}+e^{tB}e^{tA})$}
For bounded <matrices>, differentiating the symmetrized product gives $F'-(A+B)F=\tfrac12([e^{tB},A]e^{tA}+[e^{tA},B]e^{tB})$. The <variation-of-constants formula> turns this into an integral expression for $F-e^{t(A+B)}$. This retains the noncommutativity of the factors explicitly; cancellation of their leading opposite <commutators> explains why the averaged product is more accurate than either ordering alone.
Back to article page
