Grassmannian, symmetric, idempotent matrices of trace $n$?

Question

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?

Answer 1

An idenpotent symmetric matrix of trace $n$ is nothing else but an orthogonal projector onto a subspace of dimension $n$. This gives a bijection between these matrices, and $n$ dimensional subspaces of $\bf R^m$, i.e. the Grassmanian.