Let $n\in\mathbb N$, and regard $GL(n,\mathbb C)$ as a real Lie group with its usual smooth manifold structure as an open subset of $M(n,\mathbb C)$. If $\gamma:(\mathbb R,+)\to GL(n,\mathbb C)$ is a smooth [group homomorphism](/page/Group%20Homomorphism), then there exists a unique matrix $X\in M(n,\mathbb C)$ such that
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\begin{align*}
\gamma(t)=\exp(tX)
\end{align*}
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for every $t\in\mathbb R$. This matrix is
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\begin{align*}
X=\gamma'(0),
\end{align*}
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where $T_I GL(n,\mathbb C)$ is identified with $M(n,\mathbb C)$ through the open inclusion $GL(n,\mathbb C)\subset M(n,\mathbb C)$.