Editing One-parameter group (section)

==In GL(n,C)==
{{see also|Stone's theorem on one-parameter unitary groups}}
An important example in the theory of Lie groups arises when <math>G</math> is taken to be <math>\mathrm{GL}(n;\mathbb C)</math>, the group of invertible <math>n\times n</math> matrices with complex entries. In that case, a basic result is the following:<ref>{{harvnb|Hall|2015}} Theorem 2.14</ref>
:'''Theorem''': Suppose <math>\varphi : \mathbb{R} \rightarrow\mathrm{GL}(n;\mathbb C)</math> is a one-parameter group. Then there exists a unique <math>n\times n</math> matrix <math>X</math> such that
::<math>\varphi(t)=e^{tX}</math>
:for all <math>t\in\mathbb R</math>.
It follows from this result that <math>\varphi</math> is differentiable, even though this was not an assumption of the theorem. The matrix <math>X</math> can then be recovered from <math>\varphi</math> as
:<math>\left.\frac{d\varphi(t)}{dt}\right|_{t=0} = \left.\frac{d}{dt}\right|_{t=0}e^{tX}=\left.(Xe^{tX})\right|_{t=0} = Xe^0=X</math>.
This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.<ref>{{harvnb|Hall|2015}} Corollary 3.50</ref>