Comment on “condition number of Drazin inverse and their condition numbers of singular linear systems”

In this paper, we establish the condition number of Drazin inverse of a singular matrix A, where R(Ak)=R(Ak*) and k = index(A), by the Schur decomposition. Based on this form, the spectral norm and Frobenius norm of relative condition number for the Drazin inverse and level-2 condition number of the Drazin inverse are characterized. The sensitivity for the Drazin-inverse solution of singular systems is also presented. We obtain several formulas for the condition number of the Drazin inverse by spectral norm instead of the P-norm, where P is a transformation matrix of the Jordan canonical form of A, thereby improving the earlier work [Y. Wei, G. Wang, D. Wang, Condition number of Drazin inverse and their condition numbers of singular linear systems, Appl. Math. Comput. 146 (2003) 455-467].