(Complex) angle between vectors in $\mathbb{C}^n$?

Question

For real vector spaces, the Cauchy-Schwarz inequality $|x\cdot y|\leq\Vert x\Vert \Vert y\Vert$ allows one to define a unique angle $\theta\in[0,\pi]$ between vectors $x$ and $y$ via $\cos^{-1}\frac{x\cdot y}{\Vert x\Vert \Vert y\Vert}$. But the Cauchy-Schwarz inequality is still valid for complex vectors. I was wondering if one can define the "angle" between complex vectors by the same inverse cosine formula. $x\cdot y=x^{\ast T} y$ is now complex, so the angle would become complex, but what is the "principal domain" of the angle to ensure single-valuedness of the inverse cosine function? Or does such a domain exist at all? I tried analyzing the formula $$\cos^{-1}z=-i\log[z+i(1-z^2)^{1/2}]$$ without much success.

Answer 1

Yes, the regular definition of $$cos\ \theta = \frac{\overrightarrow{x}\cdot\overrightarrow{y^*}}{\|x\| \cdot \|y\|}$$ works for complex numbers, too. The principal domain = $[0, 2\pi]$.