$GL_n(C)$ is isomorphic to a lie subgroup of $GL_{2n}(R)$.
I see some posts concerning this (not the same claim):
$GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
$GL_n\mathbb{C}$ as subgroup of $GL_{2n}\mathbb{R}$
$GL(n, \mathbb{C})$ is a properly embedded Lie subgroup of $GL(2n, \mathbb{R})$
My definitions of a lie group and a ie subgroup are as follows:
Lie group is a manifold $G$ together with a binary operation $m : G×G→G$ that is a smooth map converting $G$ into a group, so that the map $G→G$ carrying $x ∈ G$ to $x^{−1}$, is smooth.
A Lie subgroup of a Lie group $G$ is a subgroup $H$ that is an immersed submanifold.
A subset $M$ of a manifold $N$ is an immersed submanifold if $M$ is endowed with a structure of manifold such that the embedding $M → N$ is an immersion.
A smooth map $f : M → N$ is called immersion if, for any $x ∈M$, the tangent map $T_x(f) : T_xM → T_f(x)N$ is injective.
Can you kindly explain-provide details for this claim.
A non-zero complex number $z = x+iy$ can be mapped to a $2\times 2$ real matrix $$ A_z = \left( \begin{matrix} x & -y \\ y & x \end{matrix} \right). $$ Since $\det A_z = x^2+y^2 \neq 0$, then $A_z \in \mathrm{GL}(2,\mathbb R)$. This establishes a correspondence $\mathrm{GL}(1,\mathbb C) \to \mathrm{GL}(2,\mathbb R)$. Can you generalize it?