Condition numbers and perturbation of the weighted Moore–Penrose inverse and weighted linear least squares problem

摘要

It is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems. 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. These explicit expressions extend the earlier work of several authors. As we know, the computed weighted least squares solution x of minx∥Ax−b∥M, in the presence of the round-off error, satisfies the perturbed equation miny∥(A+E)y−(b+f)∥M. Finally, an upper bound of ∥y−x∥N is derived for the case where the weighted matrix and vector norms and the assumption rank(A+E)=rank(A) are used.