Validated Bounds on Basis Vectors for the Null Space of a Full Rank Rectangular Matrix

An orthogonal basis for the null space of a rectangular m by n matrix, with m < n, is required in various contexts, and numerous well-known techniques, such as QR factorizations or singular value decompositions, are effective at obtaining numerical approximations to such a basis. However, validated bounds on the components of each of these null space basis vectors are sometimes required. In this note, we present a simple method for reliably computing such bounds, given an approximation to the null space. We have implemented the method, illustrating its practicality.