On optimal backward perturbation bounds for the linear least squares problem

Consider the linear least squares problem min(x) parallel to b-Ax parallel to(2), where A is an m x n (m > n) matrix, and b is an n-dimensional vector. Let y be an n-dimensional vector, and let eta(LS)(y) be the optimal backward perturbation bound defined by eta(LS)(y) = inf{parallel to F parallel to F : y is a solution to min(x) parallel to b - (A + F)x parallel to 2}. An explicit expression of eta(LS)(y) (y not equal 0) has been given in [8]. However, if we define the optimal backward perturbation bounds eta(MLS)(y) by eta(MLS)(y) = inf{parallel to F parallel to F : y is the minimum 2-norm solution to min(x) parallel to b - (A + F)x parallel to 2}, it may well be asked: How to derive an explicit expression of eta(MLS)(y)? This note gives an answer. The main result is: If b not equal 0 and y not equal 0, then eta(MLS)(y) = eta(LS)(y).