Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

THE PERTURBATION FOR THE DRAZIN INVERSE

  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A representation for the Drazin inverse of an arbitrary square matrix in terms of the eigenprojection was established by Rothblum [SIAM J. Appl. Math., 31(1976) :646-648]. In this paper perturbation results based on the representation for the Darzin inverse $A^D\;=\;(A-X)^{-1}(I-X)$ are developed. Norm estimates of $\parallel(A+E)^D-A^D\parallel_2/\parallel A^D\parallel_2$ and $\parallel(A+E)^#-A^D\parallel_2/\parallel A^D\parallel_2$ are derived when IIEI12 is small.

Keywords

References

  1. U. G. Rothblum, A representation of the Drazin inverse and characterizations of the index, SIAM J. Appl. Math. 31 (1976), 646-648. https://doi.org/10.1137/0131057
  2. M. P. Darzin, Pesudoinverses in associative rings and semigroup, Amer. Math. Monthly. 65 (1958), 506-514. https://doi.org/10.2307/2308576
  3. S. L. Campbell, C. D. Meyer, Jr. and N. J. Rose, Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. Appl. Math. 31 (1976), 411-425. https://doi.org/10.1137/0131035
  4. R. E. Hartwig and J. Levine, Applications of the Drazin inverse to the hill cryptographic system, Part III, Cryptologia 5 (1975), 443-464.
  5. C. D. Meyer and Jr., The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17 (1975), 443-464. https://doi.org/10.1137/1017044
  6. C. D. Meyer, Jr. and R. J. Plemmons, Convergent powers of a matrix with applications to iterative methods for singular linear systems, SIAM J. Numer. Anal. 14 (1977), 699-705. https://doi.org/10.1137/0714047
  7. U. G. Rothblum, Multiplicative Markov decision chains, Ph.D. dissertation, Stanford University, Stanford, CA, 1974.
  8. B. Simeon, C. Fuhrer and P. Rentrop, The Drazin inverse in multibody system dynamics, Numer. Math. 64 (1993), 521-539. https://doi.org/10.1007/BF01388703
  9. G. R. Wang, A cramer rule for finding the solution of a class of singular equations, Linear Algebra Appl. 116 (1989), 27-34. https://doi.org/10.1016/0024-3795(89)90395-9
  10. G. W. Stewart, On the continuity of the generalizied inverse, SIAM J. Appl. Math. 17 (1969), 33-45. https://doi.org/10.1137/0117004
  11. S. L. Campbell and C. D. Meyer Jr., Continuity properties of the Drazin pesudoinverse, Linear Algebra Appl. 10 (1975), 77-83. https://doi.org/10.1016/0024-3795(75)90097-X
  12. G. Rong, The error bound of the perturbation of the Drazin inverse, Linear Algebra Appl.47 (1982), 159-168. https://doi.org/10.1016/0024-3795(82)90233-6
  13. Y. Wei and G. Wang, The perturbation theory for the Drazin inverse and its applications, Linear Algebra Appl. 258 (1997), 179-186. https://doi.org/10.1016/S0024-3795(96)00159-0
  14. Y. Wei, Perturbation bound of the Drazin inverse, Appl. Math. Comput. 125 (2002), 231-244. https://doi.org/10.1016/S0096-3003(00)00126-0
  15. Y. Wei, A characterization and representation of the Drazin inverse, SIAM J. Matrix Anal. Appl. 17 (1996), 744-747. https://doi.org/10.1137/S0895479895280697
  16. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Wiley,New York, 1974.