The Stable Perturbation of the Drazin Inverse of the Square Matrices
The Stable Perturbation of the Drazin Inverse of the Square Matrices
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DOI：
10.1137/080741793
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
Qingxiang Xu;Chuanning Song;Yimin Wei
中科院分区：
文献类型：
--
作者：
Qingxiang Xu;Chuanning Song;Yimin Wei
For any $n\times n$ complex matrix $A$, let $A^D$ and $A^\pi$ be the Drazin inverse and the spectral projector of $A$, respectively, where $A^\pi=I-AA^D$. When $A$ is singular, an $n\times n$ complex matrix $B$ is said to be a stable perturbation of $A$ if $I-A^\pi-B^\pi$ is nonsingular or, equivalently, if the matrix $B$ satisfies condition (${\cal C}_s$) recently introduced by Castro-Gonzalez, Robles, and Velez-Cerrada [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 882-897]. In the perturbation analysis of the Drazin inverse, the condition of $\Vert B-A\Vert$ being small is usually implicitly assumed in the literature. In this case, the condition of $B$ being a stable perturbation of $A$ is necessary in order to ensure the continuity of the Drazin inverse. In this paper, only under the condition that $B$ is a stable perturbation of $A$, two explicit formulas for the Drazin inverse $B^D$ and the spectral projector $B^\pi$ are provided, respectively, and some upper bounds for $\Vert B^D-A^D\Vert/\Vert A^D\Vert$ and $\Vert B^\pi-A^\pi\Vert$ are derived from these formulas under certain conditions. In the case where $\Vert B-A\Vert$ is small, numerical examples are given indicating the sharpness of these norm upper bounds. Furthermore, a numerical example is provided to illustrate that a perturbation analysis of the Drazin inverse can also be conducted, even if $\Vert B-A\Vert$ is not small.
