Condition Number for Weighted Linear Least Squares Problem (Journal of Computational Mathematics, 2007)
In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem: $\min_x \|Ax-b\|_2$, where $A$ is an $m$-by-$n$ ($m \ge n$) rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm $\|\left[AT, \beta b\right] \|_F$ of the data $A$ and $b$, where $T$ is a positive diagonal matrix and $\beta$ 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.
Keywords: Moore-Penrose inverse; Condition number; Linear least squares.