$X$ is a matrix. Let $v$ be an eigenvector of $X$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$

If $X$ is diagonalizable, then we can start writing out terms using Taylor expansion of $e^{X}$ but I can't seem to get anywhere.

Thanks for the help

*Edit:* Corrected question to read '*Let* $v$ *be an eigenvector of* $X$' instead of '*Let* $v$ *be an eigenvector of* $e^X$'.