Acute perturbation of Drazin inverse and oblique projectors

Abstract

For an n×n complex matrix A with ind(A) = r; let A^D and A^{\pi } = I-AA^D be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) =s; it is said to be a stable perturbation of A; if I-{(B^{\pi }-A^{\pi })}^{\mathrm{2}} is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs) R(B^s)\cap N(A^r)=\left\{0\right\} and N(B^s)\cap R(A^r)=\left\{0\right\}, introduced by Castro-Gonz

Key words： Drazin inverse; acute perturbation; stable perturbation; spectral radius; spectral norm; oblique projection

References

1 Baksalary O, Trenkler G. Core inverse of matrices. Linear Multilinear Algebra, 2010, 58: 681–697

2 Baksalary O, Trenkler G. On column and null spaces of functions of a pair of oblique projectors. Linear Multilinear Algebra, 2013, 61: 1116–1129

3 Ben-Israel A, Greville T N E. Generalized Inverses Theory and Applications. 2nd ed. New York: Springer, 2003

4 Böttcher A, Spitkovsky I. A gentle guide to the basics of two projections theory. Linear Algebra Appl, 2010, 432: 1412–1459

5 Campbell S, Meyer C. Generalized Inverses of Linear Transformations. Philadelphia: SIAM, 2009

6 Castro González N, Koliha J, Straskraba I. Perturbation of the Drazin inverse. Soochow J Math, 2001, 27: 201–211

7 Castro González N, Koliha J, Wei Y. Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero. Linear Algebra Appl, 2000, 312: 181–189

8 Castro-González N, Martínez-Serrano M, Robles J. An extension of the perturbation analysis for the Drazin inverse. Electron J Linear Algebra, 2011, 22: 539–556

9 Castro-González N, Robles J, Vélez-Cerrada J. Characterizations of a class of matrices and perturbation of the Drazin inverse. SIAM J Matrix Anal Appl, 2008, 30: 882–897

10 Castro-González N, Vélez-Cerrada J. On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces. J Math Anal Appl, 2008, 341: 1213–1223

11 Deng C, Du H. The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces. Proc Amer Math Soc, 2006, 134: 3309–3317

12 Eagambaram M, Manjunatha Prasad K, Mohana K. Column space decomposition and partial order on matrices. Electronic J Linear Algebra, 2013, 26: 795–815

13 Galántai A. Projectors and Projection Methods. New York: Springer, 2004

14 Groβ J, Trenkler G. On the product of oblique projectors. Linear Multilinear Algebra, 1998, 44: 247–259

15 Hanke M. Iterative consistency: a concept for the solution of singular systems of linear equations. SIAM J Matrix Anal Appl, 1994, 15: 569–577

16 Hartwig R, Wang G, Wei Y. Some additive results on Drazin inverse. Linear Algebra Appl, 2001, 322: 207–217

17 Ipsen I, Meyer C. The angle between complementary subspaces. Amer Math Monthly, 1995, 102: 904–911

18 Kato T. Perturbation Theory for Linear Operators. Berlin: Springer-Verlag Berlin, 1980

19 Kirkland S, Neumann M. Group Inverses of M-Matrices and their Applications. Boca Raton: CRC Press, 2012

20 Kirkland S, Neumann M, Sze N. On optimal condition numbers for Markov chains. Numer Math, 2008, 110: 521–537

21 Koliha J. Error bounds for a general perturbation of the Drazin inverse. Appl Math Comput, 2002, 126: 181–185

22 Koliha J, Rakovcevic V, Straskraba I. The difference and sum of projectors. Linear Algebra Appl, 2004, 388: 279–288

23 Li X, Wei Y. An improvement on the perturbation of the group inverse and oblique projection. Linear Algebra Appl, 2001, 388: 53–66

24 Li X, Wei Y. A note on the perturbation bound of the Drazin inverse. Appl Math Comput, 2003, 140: 329–340

25 Liu X, Wang W, Wei Y. Continuity properties of the f1g-inverse and perturbation bounds for the Drazin inverse. Linear Algebra Appl, 2008, 429: 1026–1037

26 Liu X, Wu S, Yu Y. The perturbation of the Drazin inverse. Int J Comput Math, 2012, 89: 711–726

27 Malik S, Thome N. On a new generalized inverse for matrices of an arbitrary index. Appl Math Comput, 2014, 226: 575–580

28 Meyer C. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Review, 1975, 17: 443–464

29 Meyer C. The condition of a finite Markov chain and perturbation bounds for the limiting probabilities. SIAM J on Algebraic and Discrete Methods, 1980, 1: 273–283

30 Rice J. A theory of condition. SIAM J Numer Anal, 1966, 3: 287–310

31 Rong G. The error bound of the perturbation of the Drazin inverse. Linear Algebra Appl, 1982, 47: 159–168

32 Song C, Wei Y, Xu Q. A note on perturbation estimations for spectral projectors. Numer Funct Anal Optim, 2019 (to appear)

33 Stewart G. On the perturbation of pseudo-inverse, projections and linear least squares problems. SIAM Review, 1977, 19: 634–662

34 Stewart G. On the numerical analysis of oblique projectors. SIAM J Matrix Anal Appl, 2011, 32: 309–348

35 Sun J. On the sensitivity of the spectral projection. Linear Algebra Appl, 2005, 395: 83–94

36 Szyld D. The many proofs of an identity on the norm of oblique projections. Numer Algorithms, 2006, 42: 309–323

37 Vélez-Cerrada J, Robles J, Castro-González N. Error bounds for the perturbation of the Drazin inverse under some geometrical conditions. Appl Math Comput, 2009, 215: 2154–2161

38 Wang G, Wei Y, Qiao S. Generalized Inverses: Theory and Computations. Developments in Mathematics, Vol 53. Singapore/Beijing: Springer/Science Press, 2018

39 Wedin P. Perturbation theory for pseudo-inverses. BIT, 1973, 13: 217–232

40 Wei Y. A characterization and representation of the Drazin inverse. SIAM J Matrix Anal Appl, 1996, 17: 744–747

41 Wei Y. On the perturbation of the group inverse and oblique projection. Appl Math Comput, 1999, 98: 29–42

42 Wei Y. The Drazin inverse of updating of a square matrix with application to perturbation formula. Appl Math Comput, 2000, 108: 77–83

43 Wei Y. Perturbation bound of the Drazin inverse. Appl Math Comput, 2002, 125: 231–244

44 Wei Y. Generalized inverses of matrices. In: Hogben L, ed. Handbook of Linear Algebra. 2nd ed. Boca Raton: CRC Press, 2014, Chapter 27

45 Wei Y. Acute perturbation of the group inverse. Linear Algebra Appl, 2017, 534: 135–157

46 Wei Y, Deng C. A note on additive results for the Drazin inverse. Linear Multilinear Algebra, 2011, 59: 1319–1329

47 Wei Y, Li X. An improvement on perturbation bounds for the Drazin inverse. Numer Linear Algebra Appl, 2003, 10: 563–575

48 Wei Y, Li X, Bu F. A perturbation bound of the Drazin inverse of a matrix by separation of simple invariant subspaces. SIAM J Matrix Anal Appl, 2005, 27: 72–81

49 Wei Y, Li X, Bu F, Zhang F. Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices-application of perturbation theory for simple invariant subspaces. Linear Algebra Appl, 2006, 419: 765–771

50 Wei Y,Wang G. The perturbation theory for the Drazin inverse and its application. Linear Algebra Appl, 1997, 258: 179–186

51 Wei Y, Wu H. Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index. J Comput Appl Math, 2000, 114: 305–318

52 Wei Y, Wu H. The perturbation of the Drazin inverse and oblique projection. Appl Math Lett, 2000, 13: 77–83

53 Wei Y, Wu H. Challenging problems on the perturbation of Drazin inverse. Ann Oper Res, 2001, 103: 371–378

54 Xu Q, Song C, Wei Y. The stable perturbation of the Drazin inverse of the square matrices. SIAM J Matrix Anal Appl, 2010, 31: 1507-1520

55 Xue Y, Chen G. Perturbation analysis for the Drazin inverse under stable perturbation in Banach space. Missouri J Math Sci, 2007, 19(2): Article 3 http://www.math-cs.ucmo.edu/~mjms/2007-2p.html