# American Institute of Mathematical Sciences

December  2018, 8(4): 493-503. doi: 10.3934/naco.2018031

## Further results on the perturbation estimations for the Drazin inverse

 School of Mathematical Science, Harbin Normal University, Harbin 150025, China

* Corresponding author: H. Ma

Received  February 2018 Revised  March 2018 Published  September 2018

Fund Project: H. Ma is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232)
X. Gao is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232).

For
 $n× n$
complex singular matrix
 $A$
with ind
 $(A) = k>1$
, let
 $A^D$
be the Drazin inverse of
 $A$
. If a matrix
 $B = A+E$
with ind
 $(B) = 1$
is said to be an acute perturbation of
 $A$
, if
 $\|E\|$
is small and the spectral radius of
 $B_gB- A^DA$
satisfies
 $ρ(B_gB- A^DA) < 1,$
where
 $B_g$
is the group inverse of
 $B$
.
The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix
 $B$
satisfies geometrical condition:
 ${\mathcal R}(B) \cap {\mathcal N}(A^k) = \{ {\bf 0} \}, {\mathcal N}(B)\cap {\mathcal R}(A^k) = \{ {\bf 0} \}$
which introduced by Vélez-Cerrada, Robles, and Castro-González, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161).
Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind
 $(B) = 1$
by Wei (Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157).
Citation: Haifeng Ma, Xiaoshuang Gao. Further results on the perturbation estimations for the Drazin inverse. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 493-503. doi: 10.3934/naco.2018031
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