The weighted Drazin inverse of perturbed matrices with related support idempotents

This paper deals with the perturbation of the W-weighted Drazin inverse, AD,W, of rectangular matrices with related support idempotents, Aσ,W. We characterize matrices B such that Bσ,WW = Aσ,WW providing an algebraic structure for B and a formula for BD,W. Similarly, we obtain characterizations of rectangular matrices B such that WBσ,W = WAσ,W. We show two classes of perturbed matrices to which these results can be applied. Further, we derive upper bounds for ∥BD,W∥ and ∥BD,W − AD,W∥/∥AD,W∥. Finally, we consider an application of our results to the perturbation of linear systems.