Drazin inverse

A Drazin inverse of an operator $A$ is an operator, $B$ , such that

AB=BA,

BAB=B,

ABA=A-U,

where the spectral radius $r(U)=0$ . The Drazin inverse ( $B$ ) is denoted by $A^{D}$ . It exists, if $0$ is not an accumulation point of $\sigma(A)$ .

For example, a projection operator is its own Drazin inverse, $P^{D}=P$ , as $PPP=PP=P$ ; for a Shift operator $S^{D}=0$ holds.

The following are some other useful properties of the Drazin inverse:

1.

$(A^{D})^{*}=(A^{*})^{D}$ ;
2.

$A^{D}=(A+\alpha P^{(A)})^{-1}(I-P^{(A)})$ , where $P^{(A)}:=I-A^{D}A$ is the spectral projection of $A$ at $0$ and $\alpha\neq 0$ ;
3.

$A^{\dagger}=(A^{*}A)^{D}A^{*}=A^{*}(AA^{*})^{D}$ , where $A^{\dagger}$ is the Moore-Penrose pseudoinverse of $A$ ;
4.

$A^{D}=A^{m}(A^{2m+1})^{\dagger}A^{m}$ for $m\geq\mbox{ind}(A)$ , if $\mbox{ind}(A):=\min\{k:\operatorname{Im}A^{k}=\operatorname{Im}A^{k+1}\}$ is finite;
5.

If the matrix is represented explicitly by its Jordan canonical form, ( $\Lambda$ regular and $N$ nilpotent), then

$\left(E\begin{bmatrix}\Lambda&0\\ 0&N\end{bmatrix}E^{-1}\right)^{D}=E\begin{bmatrix}\Lambda^{-1}&0\\ 0&0\end{bmatrix}E^{-1};$
6.

Let $e_{\lambda}^{A}$ denote an eigenvector of $A$ to the eigenvalue $\lambda$ . Then $e_{\lambda}^{A}+t(\lambda I-A)^{D}he_{\lambda}^{A}+O(t^{2})$ is an eigenvector of $A+th$ .

Title	Drazin inverse
Canonical name	DrazinInverse
Date of creation	2013-03-22 13:58:05
Last modified on	2013-03-22 13:58:05
Owner	kronos (12218)
Last modified by	kronos (12218)
Numerical id	29
Author	kronos (12218)
Entry type	Definition
Classification	msc 47S99
Related topic	MoorePenroseGeneralizedInverse