# Bound on euclidean norm

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!

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Why did you distinguish the term $k=1$ from the sum? The expression in brackets is by the way $\mathrm e^{tM}-I$ where $I$ is the identity matrix of the dimension $n\times n$. Would you also clarify, why are you interested in such bounds? –  Ilya Aug 24 '12 at 7:38
There is no particular reason why I distinguish k=1. I thought it would be helpful, because I wan to calculate the interception point of the upper function f(t)=|...| with an onther curve. I hoped to seperate the t by distinguishing k=1. Note, the solution should be less than or equal the upper function. –  Oddler Aug 24 '12 at 8:09