On the existence of bases of class \(C^ p\) of the kernel and the image of a matrix function
This paper examines the existence of bases of class \(C^ p\) for the kernel and the range of a constant rank rectangular matrix function of q real variables defined on a domain \(\Omega \subset {\mathbb{R}}^ q\). The major technical difficulty is in getting the bases to be defined on all of \(\Omega\). When \(q=1\) and \(\Omega\) is an interval this was done by \textit{V. Doležal} [Časopis Pěst. Mat. 89, 466-469 (1964; Zbl 0187.371)]. An example is given with \(\Omega ={\mathbb{R}}^ 3-\{0\}\) that shows some restrictions on \(\Omega\) are necessary. This paper proves the existence of the bases under the assumption that \(\Omega\) is \(C^ p\) diffeomorphic to \({\mathbb{R}}^ q\) so that \(\Omega\) need not be compact. Several related issues dealing with generalized inverses and the solutions of systems of equations are also discussed.
constant rank rectangular matrix function
scientific article; zbMATH DE number 4148323
scientific article