Exponential product defect identity ($e^{tA}{e}^{tB}-e^{t(A+B)}$)

For finite matrices, differentiate the ordered product to obtain ${\Phi }^{\prime }-(A+B)\Phi =[e^{tA},B]e^{tB}$. Multiplying the error by $e^{-t(A+B)}$ and integrating gives $\Phi (t)-e^{t(A+B)}={\int }_0^t{e}^{(t-s)(A+B)}[e^{sA},B]e^{sB}ds$. This exact variation-of-constants formula identifies noncommutativity as the source of Lie-Trotter splitting error.

For symmetric negative-definite matrices, put $a={\lambda }_{\max }(A)$, $b={\lambda }_{\max }(B)$ and $c={\lambda }_{\max }(A+B)\le a+b$. The spectral theorem gives exact exponential norm bounds, and $\Vert [e^{sA},B]{\Vert }_2\le 2\Vert B{\Vert }_2{e}^{sa}$. Integration in the exponential product defect identity proves the displayed estimate if $c<a+b$. At equality its continuous limit is $2t\Vert B{\Vert }_2{e}^{tc}$; the quotient must not be interpreted as division by zero.