$ℝP^n$微分同胚于 迹1的幂等对称阵组成的$ℝ^{(n+1)^2}$子流形

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Wikipedia
The projective $n$-space is in fact diffeomorphic to the submanifold of $ℝ^{(n+1)^2}$ consisting of all symmetric $(n + 1) × (n + 1)$ matrices of trace 1 that are also idempotent linear transformations.

如何证明？

Czhang271828

A symmetric idenpotent matrix of trace $k$ is nothing but a projection onto an  $k$-dimensional space (obvious, or found in the textbook of 丘维声 (maybe)). Hence the Grassmannian $G_{n}(\mathbb F^m)$, consisting of $m$-dimensional linear subspaces of $\mathbb F^n$, is isomorphic to the manifold of symmetric idenpotent matrices in $\mathbb F^{n\times n}$ of trace $m$. Here $G_{n+1}(\mathbb F^1)=\mathbb P_{\mathbb F}^n$.