If $a_1,a_2>0$ then by the mean value theorem $$ |e^{-a_1} -e^{-a_2}|\le |a_1-a_2|\max\{e^{-a_1},e^{-a_2}\}\le|a_1-a_2|(e^{-a_1}+e^{-a_2}). $$ Is there an estimate like that for matrix exponents?
Namely, let $A_1$ and $A_2$ be symmetric positive definite $n\times n$ matrices. Does it hold $$ |e^{-A_1} -e^{-A_2}|\le C_n|A_1-A_2|(|e^{-A_1}|+|e^{-A_2}|), $$ where $|\cdot|$ is some matrix norm and $C_n$ is constant, depending on $n$ only? If no, that estimate holds here?