Vector norm such that $\| (1,i) \| = 6$

Question

Is there a norm $\| \cdot \|: \mathbb{C}^2 \rightarrow \mathbb{R}^+$ such that $\bigg\| \pmatrix{1\\i} \bigg\| = 6$?

I have come across this question and I don't really know what to look for here. Trying out random norms to see if one of them give $6$ doesn't look like a good idea, and I can't think of any properties of norms that the function in the example would not obey to prove such norm doesn't exist.

Answer 1

Consider $$\Vert x \Vert = 3\sqrt{x^H x}$$ where $x^H$ is the transpose - conjugation operation, i.e. Hermitian operator. This norm satisfies the three basic norm properties, i.e.

1. $ \Vert x \Vert = 0 $ only if $x = 0$
2. For $\alpha \in \mathbb{C}$,$ \Vert \alpha x \Vert = 3\sqrt{(\alpha x)^H \alpha x} = 3 \sqrt{ \vert \alpha \vert^2 x^H x } = \vert \alpha \vert 3\sqrt{ x^H x } = \vert \alpha \vert \Vert x \Vert$
3. $ \Vert x + y \Vert \leq \Vert x \Vert + \Vert y \Vert $

Answer 2

Hint: if $\| \cdot \|$ is a norm, then so is $c \| \cdot \|$ for any constant $c > 0$.