# 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
##### References:

show all references

##### References:
 [1] Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1 [2] Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93 [3] Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843 [4] Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975 [5] Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115 [6] Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139 [7] Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179 [8] Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232 [9] Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 [10] Rui Zou, Yongluo Cao, Gang Liao. Continuity of spectral radius over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3977-3991. doi: 10.3934/dcds.2018173 [11] Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239 [12] Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221 [13] Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015 [14] Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381 [15] Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143 [16] Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 [17] Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117 [18] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [19] Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239 [20] Daria Bugajewska, Mirosława Zima. On the spectral radius of linearly bounded operators and existence results for functional-differential equations. Conference Publications, 2003, 2003 (Special) : 147-155. doi: 10.3934/proc.2003.2003.147

Impact Factor: