I am stuck on a problem, I don't know is I am being wrong or, maybe, the text is.
Let $\phi: V\times V \to \mathbb{C}$ be a Hermitian form on a complex vector space $V$.
Find the rank and the signature of $\psi$ in the case $V = \mathbb{C}^3$ and
$$\psi(x, x) = |x_1 + ix_2|^2 + |x_2 + ix_3|^2 + |x_3 + ix_1|^2 - |x_1 + x_2 + x_3|^2$$
Show in general that, if $n > 2$, then
$$\phi(u, v) = \frac{1}{n}\sum_{k = 1}^{n} \zeta^{-k}\psi(u + \zeta^k v, u + \zeta^k v)$$
where $\zeta = e^{2\pi i/n}$
Now I think I have found the rank to be $3$, but when I have to deal with the second command (the proof), I get a mess, or anyway I don't get the result it says I should get.
Any hints?
Thank you!