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Let $M$ be the set of $(n+1)\times(n+1)$ symmetric, idempotent matrices of trace 1. What is the inverse function of $f:\mathbb{R}P^n\rightarrow M$ defined by $[x_1,\ldots,x_{n+1}]\mapsto\left(\frac{x_ix_j}{\sum x_k^{2}}\right)$?

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The inverse function you want maps each such matrix to its image (its column space, if you like), which is a $1$-dimensional subspace, that is, a point of the projective space.

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