Decaying bound for symmetric negative matrix splitting ($\Vert E(t){\Vert }_2\le 2\Vert B{\Vert }_2(e^{t(a+b)}-e^{tc})/(a+b-c)$)

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.