I came across this link on planetmath and a few facts on that link are confusing me.
According to planetmath, any $1$-parameter subgroup in $GL_n(\mathbb{C})$ arises from the exponential map. That is, for $\phi:\mathbb{R}\rightarrow GL_n(\mathbb{C})$, any one parameter subgroup is of the form $ \phi(t)=e^{tA}, $ where $A\in T_e GL_n(\mathbb{C}) \cong M_n(\mathbb{C})$.
Why is the domain for $\phi$ the reals, and not the complexes?
The same link says that the one-to-one correspondence between tangent vectors at the identity and one-parameter subgroups is established via the exponential map instead of the matrix exponential.
What do they mean by the matrix exponential?