Condition number for weighted linear least squares problem

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: minx ||Ax - b||2, where A is an m-by-n (m ≥ n) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Probenius norm ||[AT, βb]||F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.