A Note on Smooth Matrices of Constant Rank

Abstract

We show that, given a $C^h$ time–varying matrix A(t) of constant rank, there exists a $C^h$ matrix H(t) such that the rows of H(t)A(t) are an orthonormal basis of the space spanned by the rows of A(t). We present some consequences of this result and, in particular, we prove a version for $m \times n$ matrices of Doležal's Theorem. These results are not new, and references are given. All the proofs of the results stated in these references, with the exception of those based on the use of differential equations — which holds only for $h \geq 1$ —, find suitable $C^h$ matrices defined on overlapping subsets of the domain and then patch them together without losing regularity and the other required properties. In our approach the patching needs to be done only for matrices consisting of one row and all the remaining results are obtained by usual algebraic tools.