# 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. 1.

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

2. 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. 3.

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

4. 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. 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. 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 DrazinInverse 2013-03-22 13:58:05 2013-03-22 13:58:05 kronos (12218) kronos (12218) 29 kronos (12218) Definition msc 47S99 MoorePenroseGeneralizedInverse