It is well known that the set of all square invertible real matrices has two connected components. The set of all m x n rectangular real matrices of rank r has only one connected component when m not-equal n or r < m = n . We show that all these connected components are connected by analytic regular arcs. We apply this result to establish the existence of p-times differentiable bases of the kernel and the image of a rectangular real matrix function of several real variables.
Proceedings of the American Mathematical Society
Evard, J. C. and Jafari, Farhad (1994). "The Set of All MXN Rectangular Real Matrices of Rank-R Is Connected by Analytic Regular Arcs." Proceedings of the American Mathematical Society 120.2, 413-419.