Existence of real matrices acting on $\mathbb C^n$

Question

Lately I've been interested in how real operators act on complex vector spaces. Rather than restricting a vector space to a given subspace, I'm curious what happens when we place restrictions on the underlying field. In general, given any two vectors $u,v \in \mathbb C^n$, $||u||=||v||$, there is a unitary matrix $U$ such that $Uu=v$. I'm curious how these matrices behave if we restrict them to having real entries (i.e. $U$ is orthogonal rather than unitary) while still acting on the space $\mathbb C^n$. In general, given two vectors $u,v \in \mathbb C^n$ of the same norm, there will not be a real orthogonal matrix $U$ so that $Uu=v$. In this case, what are the conditions on $u$ and $v$ so that an orthogonal $U$ exists?

Answer 1

You can split your vectors into real and imaginary parts: $u=u_1+iu_2$, $v=v_2+iv_2$ with $u_1$ etc., real vectors. An orthogonal matrix takes $u$ to $v$ iff it takes $u_1$ to $v_1$ and $u_2$ to $v_2$. As it's orthogonal, it must preserve lengths and angles between vectors. So we must have $|u_1|=|v_1|$, $|u_2|=|v_2|$ and $u_1\cdot u_2=v_1\cdot v_2$. These conditions are also sufficient.