I am interested in why the functions $f_{ij} \colon \mathbb{PR}^n \to \mathbb{R}, \ [x] \mapsto x_ix_j$ define a diffeomorphism between $\mathbb{PR}^n$ and set of all symmetric matrices of dimension $n+1$ with trace $1$ and satisfying $AA = A$ and why $\mathbb{PR}^n$ is compact.
I understand everything except $AA=A$ and regarding the compactness I am not sure because I have NoPhD! :P