# 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:
 [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses Theory and Applications, Wiley, New York, 1974; 2nd edition, Springer, New York, 2003. [2] A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [3] S. L. Campbell and C. D. Meyer, Continuity properties of the Drazin pseudoinverses, Linear Algebra Appl., 10 (1975), 77-83. [4] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979; SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719048.ch0. [5] N. Castro-González, J. Robles and J. Y. Vélez-Cerrada, Characterizations of a class of matrices and perturbation of the Drazin inverse, SIAM. J. Matrix Anal. Appl., 30 (2008), 882-897. doi: 10.1137/060653366. [6] N. Castro-González and J. Y. Vélez-Cerrada, On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1213-1223. doi: 10.1016/j.jmaa.2007.10.066. [7] N. Castro-González, M. F. Martínez-Serrano and J. Robles, An extension of the perturbation analysis for the Drazin inverse, Electron. J. Linear Algebra, 22 (2011), 539-556. doi: 10.13001/1081-3810.1456. [8] D. S. Cvetković-Ilić and Y. Wei, Algebraic Properties of Generalized Inverses, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6349-7. [9] M. Eierman, I. Marek and W. Niethammer, On the solution of singular linear systems of algebraic equations by semi-iterative methods, Numer. Math., 53 (1988), 265-283. doi: 10.1007/BF01404464. [10] A. Galántai, Projectors and Projection Methods, Springer, New York, 2004. [11] R. A. Horn and C. R. Johnson, Matrix Analysis, Second Edition, Cambridge University Press, Cambridge, 2013. [12] J. Ji and Y. Wei, The Drazin inverse of an even-order tensor and its application to singular tensor equations, Comput. Math. Appl., 75 (2018), 3402-3413. doi: 10.1016/j.camwa.2018.02.006. [13] S. Kirkland and M. Neumann, Group Inverses of M-Matrices and their Applications, CRC Press, 2012. [14] J. J. Koliha, Error bounds for a general perturbation of the Drazin inverse, Appl. Math. Comput., 126 (2002), 181-185. doi: 10.1016/S0096-3003(00)00149-1. [15] X. Li and Y. Wei, An improvement on the perturbation of the group inverse and oblique projection, Linear Algebra Appl., 338 (2001), 53-66. doi: 10.1016/S0024-3795(01)00369-X. [16] X. Li and Y. Wei, A note on the perturbation bound of the Drazin inverse, Appl. Math. Comput., 140 (2003), 329-340. doi: 10.1016/S0096-3003(02)00230-8. [17] H. Ma, Acute perturbation bounds of weighted Moore-Penrose inverse, Int. J. Comput. Math., 95 (2018), 710-720. doi: 10.1080/00207160.2017.1294689. [18] C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Review, 17 (1975), 443-464. doi: 10.1137/1017044. [19] G. Rong, The error bound of the perturbation of the Drazin inverse, Linear Algebra Appl., 47 (1982), 159-168. doi: 10.1016/0024-3795(82)90233-6. [20] A. Sidi and Y. Kanevsky, Orthogonal polynomials and semi-iterative methods for the Drazin-inverse solution of singular linear systems, Numer. Math., 93 (2003), 563-581. doi: 10.1007/s002110100379. [21] G. W. Stewart, On the perturbation of pseudo-inverse, projections and linear least squares problems, SIAM Review, 19 (1977), 634-662. doi: 10.1137/1019104. [22] G. W. Stewart, On the numerical analysis of oblique projectors, SIAM J. Matrix Anal. Appl., 32 (2011), 309-348. doi: 10.1137/100792093. [23] D. Szyld, Equivalence of convergence conditions for iterative methods for singular equations, Numer. Linear Algebra Appl., 1 (1994), 151-154. doi: 10.1002/nla.1680010206. [24] D. Szyld, The many proofs of an identity on the norm of oblique projections, Numer. Algorithms, 42 (2006), 309-323. doi: 10.1007/s11075-006-9046-2. [25] J. Y. Vélez-Cerrada, J. Robles and N. Castro-González, Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161. doi: 10.1016/j.amc.2009.08.003. [26] P. Å. Wedin, Perturbation theory for pseudo-inverses, BIT, 13 (1973), 217-232. [27] Y. Wei, Expressions for the Drazin inverse of a 2 × 2 block matrix, Linear Multilinear Algebra, 45 (1998), 131-146. doi: 10.1080/03081089808818583. [28] Y. Wei, On the perturbation of the group inverse and oblique projection, Appl. Math. Comput., 98 (1999), 29-42. doi: 10.1016/S0096-3003(97)10151-5. [29] Y. Wei, Perturbation bound of the Drazin inverse, Appl. Math. Comput., 125 (2002), 231-244. doi: 10.1016/S0096-3003(00)00126-0. [30] Y. Wei, Generalized inverses of matrices, Chapter 27 of Handbook of Linear Algebra, Edited by Leslie Hogben, Second edition, CRC Press, Boca Raton, FL, 2014. [31] Y. Wei, Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157. doi: 10.1016/j.laa.2017.08.009. [32] Y. Wei and X. Li, An improvement on perturbation bounds for the Drazin inverse, Numer. Linear Algebra Appl., 10 (2003), 563-575. doi: 10.1002/nla.336. [33] Y. Wei, X. Li, F. Bu and F. Zhang, Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices-application of perturbation theory for simple invariant subspaces, Linear Algebra Appl., 419 (2006), 765-771. doi: 10.1016/j.laa.2006.06.015. [34] Y. Wei, X. Li and F. Bu, A perturbation bound of the Drazin inverse of a matrix by separation of simple invariant subspaces, SIAM J. Matrix Anal. Appl., 27 (2005), 72-81. doi: 10.1137/S0895479804439948. [35] Y. Wei and H. Wu, The perturbation of the Drazin inverse and oblique projection, Appl. Math. Lett., 13 (2000), 77-83. doi: 10.1016/S0893-9659(99)00189-5. [36] Y. Wei and H. Wu, Challenging problems on the perturbation of Drazin inverse, Ann. Oper. Res., 103 (2001), 371-378. doi: 10.1023/A:1012993626289. [37] Q. Xu, C. Song and Y. Wei, The stable perturbation of the Drazin inverse of the square matrices, SIAM J. Matrix Anal. Appl., 31 (2010), 1507-1520. doi: 10.1137/080741793.

show all references

##### References:
 [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses Theory and Applications, Wiley, New York, 1974; 2nd edition, Springer, New York, 2003. [2] A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [3] S. L. Campbell and C. D. Meyer, Continuity properties of the Drazin pseudoinverses, Linear Algebra Appl., 10 (1975), 77-83. [4] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979; SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719048.ch0. [5] N. Castro-González, J. Robles and J. Y. Vélez-Cerrada, Characterizations of a class of matrices and perturbation of the Drazin inverse, SIAM. J. Matrix Anal. Appl., 30 (2008), 882-897. doi: 10.1137/060653366. [6] N. Castro-González and J. Y. Vélez-Cerrada, On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1213-1223. doi: 10.1016/j.jmaa.2007.10.066. [7] N. Castro-González, M. F. Martínez-Serrano and J. Robles, An extension of the perturbation analysis for the Drazin inverse, Electron. J. Linear Algebra, 22 (2011), 539-556. doi: 10.13001/1081-3810.1456. [8] D. S. Cvetković-Ilić and Y. Wei, Algebraic Properties of Generalized Inverses, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6349-7. [9] M. Eierman, I. Marek and W. Niethammer, On the solution of singular linear systems of algebraic equations by semi-iterative methods, Numer. Math., 53 (1988), 265-283. doi: 10.1007/BF01404464. [10] A. Galántai, Projectors and Projection Methods, Springer, New York, 2004. [11] R. A. Horn and C. R. Johnson, Matrix Analysis, Second Edition, Cambridge University Press, Cambridge, 2013. [12] J. Ji and Y. Wei, The Drazin inverse of an even-order tensor and its application to singular tensor equations, Comput. Math. Appl., 75 (2018), 3402-3413. doi: 10.1016/j.camwa.2018.02.006. [13] S. Kirkland and M. Neumann, Group Inverses of M-Matrices and their Applications, CRC Press, 2012. [14] J. J. Koliha, Error bounds for a general perturbation of the Drazin inverse, Appl. Math. Comput., 126 (2002), 181-185. doi: 10.1016/S0096-3003(00)00149-1. [15] X. Li and Y. Wei, An improvement on the perturbation of the group inverse and oblique projection, Linear Algebra Appl., 338 (2001), 53-66. doi: 10.1016/S0024-3795(01)00369-X. [16] X. Li and Y. Wei, A note on the perturbation bound of the Drazin inverse, Appl. Math. Comput., 140 (2003), 329-340. doi: 10.1016/S0096-3003(02)00230-8. [17] H. Ma, Acute perturbation bounds of weighted Moore-Penrose inverse, Int. J. Comput. Math., 95 (2018), 710-720. doi: 10.1080/00207160.2017.1294689. [18] C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Review, 17 (1975), 443-464. doi: 10.1137/1017044. [19] G. Rong, The error bound of the perturbation of the Drazin inverse, Linear Algebra Appl., 47 (1982), 159-168. doi: 10.1016/0024-3795(82)90233-6. [20] A. Sidi and Y. Kanevsky, Orthogonal polynomials and semi-iterative methods for the Drazin-inverse solution of singular linear systems, Numer. Math., 93 (2003), 563-581. doi: 10.1007/s002110100379. [21] G. W. Stewart, On the perturbation of pseudo-inverse, projections and linear least squares problems, SIAM Review, 19 (1977), 634-662. doi: 10.1137/1019104. [22] G. W. Stewart, On the numerical analysis of oblique projectors, SIAM J. Matrix Anal. Appl., 32 (2011), 309-348. doi: 10.1137/100792093. [23] D. Szyld, Equivalence of convergence conditions for iterative methods for singular equations, Numer. Linear Algebra Appl., 1 (1994), 151-154. doi: 10.1002/nla.1680010206. [24] D. Szyld, The many proofs of an identity on the norm of oblique projections, Numer. Algorithms, 42 (2006), 309-323. doi: 10.1007/s11075-006-9046-2. [25] J. Y. Vélez-Cerrada, J. Robles and N. Castro-González, Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161. doi: 10.1016/j.amc.2009.08.003. [26] P. Å. Wedin, Perturbation theory for pseudo-inverses, BIT, 13 (1973), 217-232. [27] Y. Wei, Expressions for the Drazin inverse of a 2 × 2 block matrix, Linear Multilinear Algebra, 45 (1998), 131-146. doi: 10.1080/03081089808818583. [28] Y. Wei, On the perturbation of the group inverse and oblique projection, Appl. Math. Comput., 98 (1999), 29-42. doi: 10.1016/S0096-3003(97)10151-5. [29] Y. Wei, Perturbation bound of the Drazin inverse, Appl. Math. Comput., 125 (2002), 231-244. doi: 10.1016/S0096-3003(00)00126-0. [30] Y. Wei, Generalized inverses of matrices, Chapter 27 of Handbook of Linear Algebra, Edited by Leslie Hogben, Second edition, CRC Press, Boca Raton, FL, 2014. [31] Y. Wei, Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157. doi: 10.1016/j.laa.2017.08.009. [32] Y. Wei and X. Li, An improvement on perturbation bounds for the Drazin inverse, Numer. Linear Algebra Appl., 10 (2003), 563-575. doi: 10.1002/nla.336. [33] Y. Wei, X. Li, F. Bu and F. Zhang, Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices-application of perturbation theory for simple invariant subspaces, Linear Algebra Appl., 419 (2006), 765-771. doi: 10.1016/j.laa.2006.06.015. [34] Y. Wei, X. Li and F. Bu, A perturbation bound of the Drazin inverse of a matrix by separation of simple invariant subspaces, SIAM J. Matrix Anal. Appl., 27 (2005), 72-81. doi: 10.1137/S0895479804439948. [35] Y. Wei and H. Wu, The perturbation of the Drazin inverse and oblique projection, Appl. Math. Lett., 13 (2000), 77-83. doi: 10.1016/S0893-9659(99)00189-5. [36] Y. Wei and H. Wu, Challenging problems on the perturbation of Drazin inverse, Ann. Oper. Res., 103 (2001), 371-378. doi: 10.1023/A:1012993626289. [37] Q. Xu, C. Song and Y. Wei, The stable perturbation of the Drazin inverse of the square matrices, SIAM J. Matrix Anal. Appl., 31 (2010), 1507-1520. doi: 10.1137/080741793.
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