Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Equation Solving Generalized Inverses Read chapter Read first chapter

2018 | OriginalPaper | Chapter

8. Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse

Authors : Guorong Wang, Yimin Wei, Sanzheng Qiao

Published in: Generalized Inverses: Theory and Computations

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Let A be a given matrix. When computing a generalized inverse of A, due to rounding error, we actually obtain the generalized inverse of a perturbed matrix \(B=A+E\) of A. It is natural to ask if the generalized inverse of B is close to that of A when the perturbation E is sufficiently small.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Parallel Algorithms for Computing the Generalized Inverses
next chapter Perturbation Analysis of the Drazin Inverse and the Group Inverse
Literature
1.
2.
G.W. Stewart, On the perturbation of pseudo-inverse, projections and linear squares problems. SIAM Rev. 19, 634–662 (1977)MathSciNetCrossRef
3.
4.
G. Wang, Perturbation theory for weighted Moore-Penrose inverse. Comm. Appl. Math. Comput. 1, 48–60 (1987). in ChineseMathSciNet
5.
X. He, On the continuity of generalized inverses application of the theory of numerical dependence. Numer. Math. J. Chinese Univ. 1, 168–172 (in Chinese, 1979)
6.
7.
G. Chen, The minimizing property of the weighted condition number in the problem of matrix perturbation. J. East China Norm. Univ. 7, 1–7 (1992). in Chinese
8.
Y. Wei, D. Wang, Condition numbers and perturbation of the weighted Moore-Penrose inverse and weighted linear least squares problem. Appl. Math. Comput. 145, 45–58 (2003)MathSciNetMATH
9.
R.E. Hartwig, Singular value decomposition and the Moore-Penrose inverse of bordered matices. SIAM J. Appl. Math. 31, 31–41 (1976)MathSciNetCrossRef
10.
Y. Wei, H. Wu, Expression for the perturbation of the weighted Moore-Penrose inverse. Comput. Math. Appl. 39, 13–18 (2000)MathSciNetCrossRef
11.
12.
G. Wang, J. Kuang, On a new measure of degree of ill-condition for matrix. Numer. Math. J. Chinese Univ. 1, 20–30 (in Chinese, 1979)
13.
14.
G. Chen, Y. Wei, Y. Xue, The generalized condition numbers of bounded linear operators in Banach spaces. J. Aust. Math. Soc. 76, 281–290 (2004)MathSciNetCrossRef
15.
G. Wang, Some necessary and sufficient condition for minimizing the condition number of a matrix. J. Shanghai Norm. Univ. 15, 10–14 (1986). in ChineseMATH
16.
H. Diao, Y. Wei, On Frobenius normwise condition numbers for Moore-Penrose inverse and linear least-squares problems. Numer. Linear Algebra Appl. 14, 603–610 (2007)MathSciNetCrossRef
17.
Y. Wei, W. Xu, S. Qiao, H. Diao, Componentwise condition numbers for generalized matrix inversion and linear least squares. Numer. Math. J. Chinese Univ. (Engl. Ser.) 14, 277–286 (2005)
18.
Y. Wei, Y. Cao, H. Xiang, A note on the componentwise perturbation bounds of matrix inverse and linear systems. Appl. Math. Comput. 169, 1221–1236 (2005)MathSciNetMATH
19.
F. Cucker, H. Diao, Y. Wei, On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems. Math. Comp. 76, 947–963 (2007)MathSciNetCrossRef
20.
H. Diao, W. Wang, Y. Wei, S. Qiao, On condition numbers for Moore-Penrose inverse and linear least squares problem involving Kronecker products. Numer. Linear Algebra Appl. 20, 44–59 (2013)MathSciNetCrossRef
21.
L. Meng, B. Zheng, The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm. Linear Algebra Appl. 432, 956–963 (2010)MathSciNetCrossRef
22.
G.W. Stewart, J. Sun, Matrix Perturbation Theory (Academic Press, New York, 1990)MATH
23.
J. Sun, Matrix Perturbation Analysis, 2nd edn. (Science Press, Beijing, in Chinese, 2001)
24.
25.
M.E. Gulliksson, X. Jin, Y. Wei, Perturbation bound for constrained and weighted least squares problem. Linear Algebra Appl. 349, 221–232 (2002)MathSciNetCrossRef
26.
M.E. Gulliksson, P.Å. Wedin, Y. Wei, Perturbation identities for regularized Tikhonov inverses and weighted pseudoinverse. BIT 40, 513–523 (2000)MathSciNetCrossRef
27.
M. Wei, Supremum and Stability of Weighted Pseudoinverses and Weighted Least Squares Problems Analysis and Computations (Nova Science Publisher Inc., Huntington, 2001)MATH
28.
Y. Wei, G. Wang, On continuity of the generalized inverse \(A_{T, S}^{(2)}\). Appl. Math. Comput. 136, 289–295 (2003)MathSciNetMATH
29.
X. Liu, W. Wang, Y. Wei, Continuity properties of the \(\{1\}\)-inverse and perturbation bounds for the Drazin inverse. Linear Algebra Appl. 429, 1026–1037 (2008)MathSciNetCrossRef
30.
31.
Y. Wei, N. Zhang, Condition number related with generalized inverse \(A_{T, S}^{(2)}\) and constrained linear systems. J. Comput. Appl. Math. 157, 57–72 (2003)MathSciNetCrossRef
32.
H. Diao, M. Qin, Y. Wei, Condition numbers for the outer inverse and constrained singular linear system. Appl. Math. Comput. 174, 588–612 (2006)MathSciNetMATH
33.
F. Cucker, H. Diao, Y. Wei, Smoothed analysis of some condition numbers. Numer. Linear Algebra Appl. 13, 71–84 (2006)MathSciNetCrossRef
34.
P. Burgisser, F. Cucker, Smoothed analysis of Moore-Penrose inversion. SIAM J. Matrix Anal. Appl. 31, 2769–2783 (2010)MathSciNetCrossRef
35.
Z. Xu, C. Gu, B. Feng, Weighted acute perturbation for two matrices. Arab. J. Sci. Eng. ASJE. Math. 35(1D), 129–143 (2010)MathSciNetMATH
36.
X. Zhang, X. Fang, C. song, Q. Xu, Representations and norm estimations for the Moore-Penrose inverse of multiplicative perturbations of matrices. Linear Multilinear Algebra 65(3), 555–571 (2017)MathSciNetCrossRef
37.
Z. Li, Q. Xu, Y. Wei, A note on stable perturbations of Moore-Penrose inverses. Numer. Linear Alegbra Appl. 20, 18–26 (2013)MathSciNetCrossRef
38.
Q. Xu, C. Song, Y. Wei, The stable perturbation of the Drazin inverse of the square matrices. SIAM J. Matrix Anal. Appl. 31(3), 1507–1520 (2009)MathSciNetCrossRef
39.
L. Lin, T.-T. Lu, Y. Wei, On level-2 condition number for the weighted Moore-Penrose inverse. Comput. Math. Appl. 55, 788–800 (2008)MathSciNetCrossRef
40.
K. Avrachenkov, M. Haviv, Perturbation of null spaces with application to the eigenvalue problem and generalized inverses. Linear Algebra Appl. 369, 1–25 (2003)MathSciNetCrossRef
41.
Z.-C. Li, H.-T. Huang, Y. Wei, A.H.-D. Cheng, Effective Condition Number for Numerical Partial Differential Equations, 2nd edn. (Science Press and Alpha Science International Ltd., Beijing, 2015)
Metadata
Title
Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse
Authors
Guorong Wang
Yimin Wei
Sanzheng Qiao
Copyright Year
2018
Publisher
Springer Singapore
Book
Generalized Inverses: Theory and Computations

Print ISBN: 978-981-13-0145-2

Electronic ISBN: 978-981-13-0146-9

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-981-13-0146-9_8

Premium Partner

    Image Credits