Is it possible to find a suitable lower bound on $\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ in the form of $|tAb|-c|b|$ with appropriate $c$?
Or are there other practical approximations on the upper equation?
Thanks!