Question:
Prove that the set of real $n\times n$ matrices with n distinct eigenvalues is a manifold.
Attempt:
I wish to show that this set is an open subset of the set of all real $n\times n$ matrices, since $\mathbb{R}^{n^2}$ is a manifold, and an open subset of a manifold is a manifold. However, I don’t know the map to use to make this possible (i.e. I don’t know what to map $Mat(n, \mathbb{R})$ or $\mathbb{R}^{n^2}$ to.
Any help would be greatly appreciated.