Drazin inverse
A Drazin inverse of an operator $A$ is an operator, $B$, such that
$$AB=BA,$$ 
$$BAB=B,$$ 
$$ABA=AU,$$ 
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 $\alpha \ne 0$;

3.
${A}^{\u2020}={({A}^{*}A)}^{D}{A}^{*}={A}^{*}{(A{A}^{*})}^{D}$, where ${A}^{\u2020}$ is the MoorePenrose pseudoinverse^{} of $A$;

4.
${A}^{D}={A}^{m}{({A}^{2m+1})}^{\u2020}{A}^{m}$ for $m\ge \text{ind}(A)$, if $\text{ind}(A):=\mathrm{min}\{k:\mathrm{Im}{A}^{k}=\mathrm{Im}{A}^{k+1}\}$ is finite;

5.
If the matrix is represented explicitly by its Jordan canonical form^{}, ($\mathrm{\Lambda}$ regular^{} and $N$ nilpotent^{}), then
$${\left(E\left[\begin{array}{cc}\hfill \mathrm{\Lambda}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill N\hfill \end{array}\right]{E}^{1}\right)}^{D}=E\left[\begin{array}{cc}\hfill {\mathrm{\Lambda}}^{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]{E}^{1};$$ 
6.
Let ${e}_{\lambda}^{A}$ denote an eigenvector^{} of $A$ to the eigenvalue^{} $\lambda $. Then ${e}_{\lambda}^{A}+t{(\lambda IA)}^{D}h{e}_{\lambda}^{A}+O({t}^{2})$ is an eigenvector of $A+th$.
Title  Drazin inverse 

Canonical name  DrazinInverse 
Date of creation  20130322 13:58:05 
Last modified on  20130322 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 