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Further results on the perturbation estimations for the Drazin inverse
School of Mathematical Science, Harbin Normal University, Harbin 150025, China |
$n× n$ |
$A$ |
$(A) = k>1$ |
$A^D$ |
$A$ |
$B = A+E$ |
$(B) = 1$ |
$A$ |
$\|E\|$ |
$B_gB- A^DA$ |
$ρ(B_gB- A^DA) < 1,$ |
$B_g$ |
$B$ |
$B$ |
${\mathcal R}(B) \cap {\mathcal N}(A^k) = \{ {\bf 0} \}, {\mathcal N}(B)\cap {\mathcal R}(A^k) = \{ {\bf 0} \}$ |
$(B) = 1$ |
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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