Some topics on weighted Moore–Penrose inverse, weighted least squares and weighted regularized Tikhonov problems

Abstract

It is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems. The classical normwise relative condition number formulas are given by Higham [Linear Algebra Appl. 214 (1995) 193]. Here, we consider the condition number formulas for the weighted Moore–Penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems Ax= b. Moreover, we consider the nearness to rank deficiency. Finally, we give some decomposition tools and their application in the regularized weighted Tikhonov problems.