A Note on the Computation of an Orthonormal Basis for the Null Space of a Matrix

A highly regarded method to obtain an orthonormal basis, $Z$, for the null space of a matrix $A^{T}$ is the $QR$ decomposition of $A$, where $Q$ is the product of Householder matrices. In several optimization contexts $A(x)$ varies continuously with $x$ and it is desirable the $Z(x)$ vary continuously also. In this note we demonstrate that the standard implementation of the $QR$ decomposition does not yield an orthonormal basis $Z(x)$ whose elements vary continuously with $x$. We suggest three possible remedies