Journal
FRONTIERS OF MATHEMATICS IN CHINA
Volume 13, Issue 6, Pages 1427-1445Publisher
HIGHER EDUCATION PRESS
DOI: 10.1007/s11464-018-0731-y
Keywords
Drazin inverse; acute perturbation; stable perturbation; spectral radius; spectral norm; oblique projection; 15A09; 65F20
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Funding
- International Cooperation Project of Shanghai Municipal Science and Technology Commission [16510711200]
- Natural Science and Engineering Research Council (NSERC) of Canada [RGPIN-2014-04252]
- National Natural Science Foundation of China [11771099]
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For an nxn complex matrix A with ind(A) = r; let A(D) and A = I-AA(D) be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an nxn complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I-(B-A)(2) is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs)N(Ar)={0} and N(Bs)R(Ar)={0}, introduced by Castro-Gonzalez, Robles, and Velez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius (B-A) < 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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