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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?

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    But viewing $GL_n(\mathbb{C})$ as a complex Lie group, do all $1$-parameter subgroups still arise as $e^{tA}$ where $A\in M_n(\mathbb{C})$ and $ t\in \mathbb{C}$ or is there a possibility for different sort of $1$-parameter subgroups to arise in the complex setting?2012-06-30

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