# Relationship between matrix ||A|| and ||e^A - I||

If $$||\cdot||$$ is a matrix norm and $$A$$ is a matrix, is there any relation between the two quantities $$||A||$$ and $$||e^A - I||$$, where $$I$$ is the identity matrix?

I am tempted to say the former is always smaller than or equal to the latter, analogous to $$x \leq e^x - 1$$ for all $$x \in \mathbb{R}$$, but my numerical experiment suggests that it is the other way: $$||A|| \geq ||e^A - I||$$. Can some one please provide some idea?

• There is no such relationship for the Euclidean norm. Consider $A=I$ and $A=-I$ which give $\le$ and $\ge$ respectively. – Peter Foreman Jan 15 at 22:40

If the norm is submultiplicative (i.e. $$\ \|AB\,\|\le \|A\,\| \|B\,\|\$$ for all $$\ A,B\$$), then you do get \begin{align} \left\|e^A-I\right\|&= \left\|\sum_{n=1}^\infty\frac{A^n}{n!} \right\|\\ &\le \sum_{n=1}^\infty\frac{\|A\|^n}{n!}\\ &=e^{\|A\|}-1\ , \end{align} but since $$\ e^x\$$ is not bounded above by any polynomial in $$\ x\$$, neither is $$\ e^{\|A\|}-1\$$ bounded above by any polynomial in $$\ \|A\|\$$.