How to prove eigenvectors of a matrix exponential are the same as those for the matrix?

Question

It's relatively easy to show that any eigenvector of an arbitrary $n \times n$ matrix $A$ is also an eigenvector of it's matrix exponential, $B = e^A$. But how does one show the reverse is true: that any eigenvector of $B = e^A$ is also an eigenvector of $A$?

Answer 1

A counterexample is $$ A=\left[\begin{array}{cc}0&-2\pi\\2\pi&0\end{array}\right],\hspace{8mm}e^A=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]. $$ Then for example $[1,0]^T$ is an eigenvector of $e^A$ but not of $A$.