= Solution
Now $B^2=\epsilon^2I$, so splitting its <matrix exponential> into even and odd powers yields hyperbolic functions:
$$
\boxed{T=e^B=I\cosh\epsilon+\frac{\sinh\epsilon}{\epsilon}B=\begin{pmatrix}\cosh\epsilon&\sinh\epsilon\\\sinh\epsilon&\cosh\epsilon\end{pmatrix}.}
$$
The limiting coefficient at $\epsilon=0$ is again one, and $T=I$. The <eigenvectors> $(1,1)$ and $(1,-1)$ have factors $e^\epsilon$ and $e^{-\epsilon}$, respectively, showing how this hyperbolic transformation differs from a rotation.
Back to article page
