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It is well known that the set of all square invertible real matrices has two connected components. The set of all      m ×  n m \times n       rectangular real matrices of rank      r r       has only one connected component when      m ≠  n m \ne n       or      r &gt; m = n r &gt; 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 p      -times differentiable bases of the kernel and the image of a rectangular real matrix function of several real variables.

The set of all 𝑚×𝑛 rectangular real matrices of rank 𝑟 is connected by analytic regular arcs