Consider the following linear ODE
$$\frac{d\vec{x}}{dt} =A \vec{x}$$ where $A$ is an invertible matrix. This is the motivation of the question.
I wondering whether $\{e^{At}\mid t\geq 0\}$ is a group? In case yes, it is a Lie group? It is a compact group?
Moreover, does it hold that
$$ cl(<A>)= \{e^{At}\mid t\geq 0\}$$ where $cl(<A>)$ is the closure of the group generated by $A$ (wrt the Euclidean topology)?
Is obviously a semigroup. The multiplicative inverse of $e^{At}$ is $e^{-(At)}=e^{A(-t)}$. The group is $$\{e^{At}\,\vert\, t\in\Bbb R\}.$$ Isn't compact because is homeomorphic to $\Bbb R$.