Comment on “Condition number of W-weighted Drazin inverse and their condition numbers of singular linear systems”

摘要

In this paper, we establish the explicit condition number formulas for the W-weighted Drazin inverse of a singular matrix A, where A∈? m×n , W∈? n×m , ?((AW) k )=?((AW) k *), ?((WA) k )=?((WA) k *), and k=max{index(AW), index(WA)}, by the Schur decomposition of A and W. The sensitivity for the W-weighted Drazin-inverse solution of singular systems is also discussed. Based on this form of Schur decomposition, the explicit condition number formulas for the W-weighted Drazin inverse are given by the spectral norm and Frobenius norm instead of the ‖?‖ P,W -norm, where P is a transformation matrix of the Jordan canonical form of AW, thereby improving the earlier work of Lei et al. (Appl. Math. Comput. 165:185–194, [2005]) and Wang et al. (Appl. Math. Comput. 162:434–446, [2005]).