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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 rectangular real matrices of rank r has only one connected component when m ¬= 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

The Set of All m × n Rectangular Real Matrices of Rank r is Connected by Analytic Regular Arcs