I'm reading a lecture notes, and I've come across this proposition:
which says that a matrix in $GL_2(\mathbb{C})$ which preserves the upper half plane of Poincaré (the subset of elements in $\mathbb{C}$ such that their imaginary parts are positive numbers) is a matrix of real coefficients modulo the action of $\mathbb{C^*}$ (In this context $GL_2(\mathbb{C})$ acts on $\mathbb{C} \cup \lbrace \infty \rbrace$ by Möbius transformation).
Question: I didn't understand the statement of this proposition: what does it mean modulo the action of $\mathbb{C^*}$?
$\mathbb{C}^*$ can be seen as a subgroup of ${\rm GL}_2(\mathbb{C})$ via the embedding $\mathbb{C}^*\hookrightarrow{\rm GL}_2(\mathbb{C})$ that sends $\lambda$ to $\lambda I_2$. This means that such a matrix is in ${\rm GL}_2(\mathbb{R})$ up to the action of $\mathbb{C}^*$ on ${\rm GL}_2(\mathbb{C})$ i.e it is of the form $\lambda A$ with $\lambda\in\mathbb{C}^*$ and $A\in{\rm GL}_n(\mathbb{R})$. This is logical since two matrices of ${\rm GL}_2(\mathbb{C})$ that are in the same orbit under the action of $\mathbb{C}^*$ give rise to the same action on $\mathcal{H}$.