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Frontiers of Mathematics in China

, Volume 13, Issue 6, pp 1427–1445 | Cite as

Acute perturbation of Drazin inverse and oblique projectors

  • Sanzheng Qiao
  • Yimin Wei
Research Article
  • 11 Downloads

Abstract

For an n×n complex matrix A with ind(A) = r; let AD and Aπ = IAAD 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–(BπAπ)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(BπAπ) < 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.

Keywords

Drazin inverse acute perturbation stable perturbation spectral radius spectral norm oblique projection 

MSC

15A09 65F20 

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Notes

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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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiChina

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