On The Belonging Of A Perturbed Vector To A Subspace From A Numerical View Point
Claudia Fassino · CINECA IRIS Institutial Research Information System (University of Genoa) · 2007
We propose a criterion in order to establish, dealing with perturbed data, when a vector v belongs to a subspace W of a vector space U from a numerical point of view. The criterion formalizes the intuitive idea that, due to the errors that aect the knowledge of the vector and the subspace, we consider the exact vector lying in the exact subspace if a “suciently small perturbation” of v belongs to a “suciently small perturbation” of W. The criterion, obviously chosen as independent of the norm of v and of the basis of W, is computationally simple to use, but it requires the choice of a threshold. Suitable values of the threshold such that the criterion adheres to the previous intuitive idea are found both when we know a basis of the exact subspace and when an orthonormal basis of the perturbed subspace is given.