  In this paper, we introduce a new geometric description of the manifolds of
matrices of fixed rank. The starting point is a geometric description of the
Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of
dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes.
The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides
it with the structure of an analytic manifold modelled on
$\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set
$\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that
the resulting manifold is an analytic principal bundle with base
$\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general
linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we
define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of
non-full rank matrices and prove that the resulting manifold is an analytic
principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times
\mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of
$\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself,
which allows a natural definition of a neighbourhood for a given matrix, this
neighbourhood being proved to possess the structure of a Lie group. Moreover,
the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology
induced by the atlas is proven to be an embedded submanifold of the matrix
space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The
proposed geometric description then results in a description of the matrix
space $\mathbb{R}^{n \times m}$, seen as the union of manifolds
$\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with
a topology for which the matrix rank is a continuous map.
