On level-2 condition number for the weighted Moore–Penrose inverse

Abstract

In this paper, we present characterizations for the level-2 condition number of the weighted Moore–Penrose inverse, i.e., cond M N ( A ) ≤ cond M N [ 2 ] ( A ) ≤ cond M N ( A ) + 1 , where cond M N ( A ) is the condition number of the weighted Moore–Penrose inverse of a rectangular matrix and cond M N [ 2 ] ( A ) is the level-2 condition number of this problem. This paper extends the result by Cucker, Diao and Wei [F. Cucker, H. Diao, Y. Wei, On the level-2 condition number for Moore–Penrose inversion, 2005, Unpublished report] and improves the results by Wei and Wang [Y. Wei, D. Wang, Condition numbers and perturbation of weighted Moore–Penrose inverse and weighted linear least squares problem, Appl. Math. Comput. 145 (2003) 45–58].