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Further results on the perturbation estimations for the Drazin inverse

  • * Corresponding author: H. Ma

    * Corresponding author: H. Ma 
H. Ma is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232)
X. Gao is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232).
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  • For $n× n$ complex singular matrix $A$ with ind$(A) = k>1$, let $A^D$ be the Drazin inverse of $A$. If a matrix $B = A+E$ with ind$(B) = 1$ is said to be an acute perturbation of $A$ , if $\|E\|$ is small and the spectral radius of $B_gB- A^DA$ satisfies

    $ρ(B_gB- A^DA) < 1,$

    where $B_g$ is the group inverse of $B$ .

    The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix $B$ satisfies geometrical condition:

    ${\mathcal R}(B) \cap {\mathcal N}(A^k) = \{ {\bf 0} \}, {\mathcal N}(B)\cap {\mathcal R}(A^k) = \{ {\bf 0} \}$

    which introduced by Vélez-Cerrada, Robles, and Castro-González, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161).

    Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind$(B) = 1$ by Wei (Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157).

    Mathematics Subject Classification: Primary: 15A09; Secondary: 65F20.

    Citation:

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