An improvement on perturbation bounds for the Drazin inverse

Abstract

SUMMARYThe Drazin inverse of a square matrix occurs in a number of applications. It is of importance to analyse the perturbation bounds for the Drazin inverse of a matrix. Let B = A + E. Under the assumption of rank(B j ) = rank(A k ), where j and k are the indices of B and A, respectively, upper bounds ofHowever, these upper bounds do not cover the perturbation bounds of the group inverse recently given by the authors as a special case.Moreover, these perturbation bounds for the Drazin inverse are too large to be practical. In this paper, we present sharper uniÿed perturbation bounds for the Drazin inverse, which are the extensions of the recent result in the case of group inverse. It solves the problem posed by Campbell and Meyer in 1975. A numerical example is given to illustrate the sharpness of the new general bounds.

An extension of the perturbation analysis for the Drazin inverse

“…Further, if ||Γ j+1,k+1 || < 1 then [4,6,11,14,16]. The upper bounds in these examples extend the perturbation analysis to a wide class of matrices.…”

“…Let E = B − A. We note that ||E|| 1 = 10 −4 , but since I + EA # and I + A # E are singular matrices we can't apply the upper bounds given in [4,14]. Since Γ 2,2 i < 1, i = 1, 2, where Γ j,k is defined as in (3.4), we can apply upper bound in Theorem 3.5 and we obtain the results that appear in Table 3.1.…”

“…Let A denote a square complex matrix and let E be a perturbation matrix. Several authors have considered the perturbation of the Drazin inverse and exhibited inequalities bounding the relative error (A + E) D − A D / A D , under specific conditions [4,6,7,11,12,14,15,16]. Other papers are concerned with the perturbation of the group inverse, which plays an important role in the theory of Markov finite chains [5,10].…”

Characterizations of a Class of Matrices and Perturbation of the Drazin Inverse

“…Taking norms Table 5.2 with the upper bounds given in [13]. Let In Table 5.3 we compare the upper bound (5.2) using the Frobenius norm with the upper bound given in [14], formula (4.1).…”

“…This latter formula was given in [15] for B = A + E, where E = AA D EAA D and E sufficiently small. The first and third authors gave in [5] [4,5,6,8,9,12,13,14,15,16].…”