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# 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 $A+th$.

## Mathematics Subject Classification

47S99*no label found*

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## Comments

## drazin, definition

for any integer m>=0 cannot hold true, because m=0 implies that T S= Identity, so the inverse exists.

## Re: drazin, definition

> for any integer m>=0 cannot hold true, because m=0 implies

> that T S= Identity, so the inverse exists.

Thanks :)