Let $n\in\mathbb N$ and let $X\in M(n,\mathbb C)$. Define

\begin{align*} \gamma_X:\mathbb R \to GL(n,\mathbb C),\qquad t\mapsto \exp(tX). \end{align*}

Then $\gamma_X$ is a smooth one-parameter subgroup of $GL(n,\mathbb C)$ and $\gamma_X'(0)=X$, where the derivative is taken in the ambient [vector space](/page/Vector%20Space) $M(n,\mathbb C)$. Moreover, if $\gamma:\mathbb R\to GL(n,\mathbb C)$ is any smooth one-parameter subgroup satisfying $\gamma'(0)=X$, then $\gamma=\gamma_X$.