Identification of Grassmannian manifolds

Question

This is a problem on Milnor's Characteristic Classes. It asks me to show that $G_n(\mathbb{R}^m)$(all $n$-planes in $\mathbb{R}^m$) is diffeomorphic to the manifold consisting of all $m\times m$ symmetric, idempotent matrices of trace $n$. I don't have any ideas about that. Could anyone give me hints?

Answer 1

An idempotent symmetric matrix of trace $n$ has eigenvalue $1$ with multiplicity $n$ and eigenvalue $0$ with multiplicity $m-n$. Its image is a $n$-dimensional subspace of $\Bbb R^m$. This gives one direction of the correspondence.

For the other it may help to note that any subspace of $\Bbb R^m$ has an orthogonal basis.