PlanetMath: Drazin inverse

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Drazin inverse

(Definition)

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

$\displaystyle A B = B A,$

$\displaystyle B A B= B,$

$\displaystyle A B A= A- U,$

. The Drazin inverse (

) is denoted by

. It exists, if 0 is not an accumulation point of $\sigma (A)$ .

For example, a projection operator is its own Drazin inverse, , as ; for a Shift operator holds.

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

;
$A^D= (A+ \alpha P^{(A)})^{-1} (I- P^{(A)})$ , where $P^{(A)}:= I-A^D A$ is the spectral projection of at 0 and $\alpha \neq 0$ ;
$A^{\dagger}= (A^* A)^D A^* = A^* (A A^* )^D$ , where $A^{\dagger}$ is the Moore-Penrose pseudoinverse of ;
$A^D= A^m (A^{2m+1})^{\dagger} A^m$ for $m \ge$ ind, if ind $(A):= \min \{ k: \operatorname{Im} A^k = \operatorname{Im} A^{k+1} \}$ is finite;
If the matrix is represented explicitly by its Jordan canonical form, ( $\Lambda$ regular and nilpotent), then

$\displaystyle \left( E \begin{bmatrix}\Lambda & 0 \\ 0 & N \end{bmatrix} E^{-1} \right)^D = E \begin{bmatrix}\Lambda^{-1} & 0 \\ 0 & 0 \end{bmatrix} E^{-1};$
Let $e_{\lambda}^A$ denote an eigenvector of to the eigenvalue $\lambda$ . Then $e_{\lambda}^A+t (\lambda I- A)^D h e_{\lambda}^A + O(t^2)$ is an eigenvector of .