Principal Bundle Structure of Matrix Manifolds

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 G r ( R k ) of linear subspaces of dimension r < k in R k , which avoids the use of equivalence classes. The set G r ( R k ) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R ( k − r ) × r . Then, we define an atlas for the set M r ( R k × r ) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base G r ( R k ) and typical fibre GL r , the general linear group of invertible matrices in R k × k . Finally, we define an atlas for the set M r ( R n × m ) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base G r ( R n ) × G r ( R m ) and typical fibre GL r . The atlas of M r ( R n × 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 M r ( R n × m ) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space R n × m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space R n × m , seen as the union of manifolds M r ( R n × m ) , as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.