I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation $\mathbb{C}^{n+1}\rightarrow\mathbb{C}^{n+1}$, inducing the automorphism $F_{A}:\mathbb{P}^{n}\rightarrow\mathbb{P}^{n}$. Let $\omega_{FS}$ denote the Fubini-Study metric on $\mathbb{P}^{n}$. The question asks
Show that $F^{*}_{A}\omega_{FS}=\omega_{FS}$ if and only if $A\in U(n+1)$.
I could do one direction: if $A\in U(n+1)$ is unitary, then the local coordinates for $F^{*}_{A}\omega_{FS}$ is something like
$\displaystyle F^{*}_{A}\omega_{FS}=\frac{1}{(1+\|Az\|^{2})^{2}}\left((1+\|Az\|^{2})A^{t}\overline{A}-A^{t}\overline{A}\overline{z}z^{t}A^{t}\overline{A}\right)$
which turns out to be $\omega_{FS}$ (unless I am mistaken in my calculations...)
How should I prove the converse?
I mean, if we see this as a metric and apply to $z$ itself, I get something like
$\displaystyle z^{t}F^{*}_{A}\omega_{FS}\overline{z}=\frac{\|Az\|^{2}}{(1+\|Az\|^{2})^2}=\frac{\|z\|^{2}}{(1+\|z\|^{2})^{2}}=z^{t}\omega_{FS}\overline{z},$
which doesn't say anything. I have tried taking the trace and the determinant and it seems like I can't get anything close to proving that it is unitary. Can anyone point references (with proofs!) or give some hints to this question? Thanks!
This seems wrong. Try any nonzero scalar multiple of a unitary matrix. Surely $F_{cA}=F_A$.