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Acute perturbation of Drazin inverse and oblique projectors

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Abstract

For an n×n complex matrix A with ind(A) = r; let AD and Aπ = I–AAD 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–(Bπ–Aπ)2 is nonsingular, equivalently, if the matrix B satisfies the condition \(\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}\) and \(\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}\), introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(Bπ–Aπ) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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 https://doi.org/www.math-cs.ucmo.edu/~mjms/2007-2p.html

    Google Scholar 

Download references

Acknowledgements

The research of Sanzheng Qiao was partly supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200) and the Natural Science and Engineering Research Council (NSERC) of Canada (Grant RGPIN-2014-04252). Partial work was finished during the visit to Shanghai Key Laboratory of Contemporary Applied Mathematics in 2018. Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200) and the National Natural Science Foundation of China (Grant No. 11771099).

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Authors and Affiliations

  1. Department of Computing and Software, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Sanzheng Qiao

  2. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Yimin Wei

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  1. Sanzheng Qiao
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  2. Yimin Wei
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Correspondence to Yimin Wei.

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Qiao, S., Wei, Y. Acute perturbation of Drazin inverse and oblique projectors. Front. Math. China 13, 1427–1445 (2018). https://doi.org/10.1007/s11464-018-0731-y

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