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I want to show that the set $G_{k,m}$ of all self-adjoint linear maps $P:\mathbb{R^m}\to\mathbb{R^m}$ of rank $k$ with $P\cdot P=P$ is a $C^\infty$ manifold of dimension $k(m-k)$.

Here is my attempt:

I was thinking if I can use the regular value theorem to show that such set is a submanifold of the manifold of all symmetric matrix. So I defined the map $f:S\to M\times\mathbb{R}$ as $f(P)=(P^2-P,tr(P)-k) $ where $S$ is the manifold of all symmetric matrix and $M$ the set of all $m\times m$ matrix. So I tried to show that $f^{-1}(0)$ is a regular level set, but it was fruitless. Any help will be appreciated.

Jacaré
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    It's worth pointing out that we can identify these maps---namely, the orthogonal projections---with their images, which are $k$-planes. Every $k$-plane defines an orthogonal projection, so we can identify $G_{k,m}$ with the space of all $k$-planes in $\Bbb R^m$. This space is called the Grassmannian of $k$-planes in $\Bbb R^m$. https://en.wikipedia.org/wiki/Grassmannian – Travis Willse Jul 18 '22 at 00:44

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