$90$-degree angle formed by $2$ non-orthogonal complex vectors?

Question

Suppose there exist 2 complex vectors, $u = (u_1, u_2, ..., u_n)$ and $v = (v_1, v_2, ..., v_n)$, in $\Bbb C^n$ such that the angle formed between $u$ and $v$ is $\pi/2$ radians, or $90^\text{o}$. (Refer to Angle between two vectors? for the formula of the angle formed by 2 complex vectors).

For example: $u = (2+i, -3+2i)$ and $v = (-1+2i, -2-3i)$ in $\Bbb C^2$.

Then, the Hermitian form of the inner product is: $u \cdot v = 0+18i$, so $$ \theta = arccos(\frac{Re(u \cdot v)}{\Vert u \Vert \Vert v \Vert}) = arccos(0) = \frac \pi 2$$

However, $u$ and $v$ are clearly NOT orthogonal, since $u \cdot v \neq 0$.

What is going on here? Is this the distinction between "perpendicular" ($90^\text{o}$) and "orthogonal" ($u \cdot v = 0$)?

Answer 1

When talking about the geometry of $\mathbb{C}^n$, often one thinks of it as $\mathbb{R}^{2n}$, with some extra algebraic structure; in this case we're really thinking of $\mathbb{C}^n$ as a $2n$-dimensional real vector space, over vectors that happen to be written as complex numbers.

In this interpretation, you can define, for instance, the angle between two complex vectors by $\cos\theta = \Re(\bar{u}^T v)/(\|u\|\|v\|)$, and you have that $1$ and $i$ are orthogonal (the angle between them is 90 degrees in the complex plane).

The post you've linked to takes this view of how to define angles in $\mathbb{C}^n$.

---

However, a second, inconsistent view of the geometry of $\mathbb{C}^n$ treats it as a complex vector space (where the base field is $\mathbb{C}$) with complex inner product (as you've done in your post). Here $1$ and $i$ are no longer orthogonal (they're not even linearly independent!), and although you can try to define a complex "angle" between vectors by analogy to the formula for real inner product spaces, it will not correspond to the usual intuition of what angles look like in $\mathbb{R}^{2n}$.

Answer 2

A "complex" inner product, an hermitian form, defined in a complex vector space can be seen like decomposed in two real "things" defined on the underlying real vector space: the real part, a usual scalar product; and the imaginary part, a symplectic form.

The scalar product gives you information about angles and length, like usual. The symplectic form is more dificult to interpret. I understand it like information about "rotation" concerning these two vectors. In other words: the real part gives you information about how one vector projects over the other; and the imaginary part gives you something like the "angular momentum" of both vectors.

For example, take $\mathbb{C}=\mathbb{R}^2$, and consider $u=1+2i$ and $v=-2+i$. You get $$ u\cdot v=0-5i $$

If you draw these vectors you obtain that they are orthogonal in the usual sense.

The problem, in fact, is of terminology, I guess: we shouldn't call orthogonal to complex vectors such that $u\cdot v=0$, but something like "totally orthogonal" or like that.