# Drazin inverse

In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix.

Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak). The Drazin inverse of A is the unique matrix AD which satisfies

${\displaystyle A^{k+1}A^{D}=A^{k},\quad A^{D}AA^{D}=A^{D},\quad AA^{D}=A^{D}A.}$
• If A is invertible with inverse ${\displaystyle A^{-1}}$, then ${\displaystyle A^{D}=A^{-1}}$.
• The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A#. The group inverse can be defined, equivalently, by the properties AA#A = A, A#AA# = A#, and AA# = A#A.
• A projection matrix P, defined as a matrix such that P2 = P, has index 1 (or 0) and has Drazin inverse PD = P.
• If A is a nilpotent matrix (for example a shift matrix), then ${\displaystyle A^{D}=0.}$

The hyper-power sequence is

${\displaystyle A_{i+1}:=A_{i}+A_{i}\left(I-AA_{i}\right);}$ for convergence notice that ${\displaystyle A_{i+j}=A_{i}\sum _{k=0}^{2^{j}-1}(I-AA_{i})^{k}.}$

For ${\displaystyle A_{0}:=\alpha A}$ or any regular ${\displaystyle A_{0}}$ with ${\displaystyle A_{0}A=AA_{0}}$ chosen such that ${\displaystyle \|A_{0}-A_{0}AA_{0}\|<\|A_{0}\|}$ the sequence tends to its Drazin inverse,

${\displaystyle A_{i}\rightarrow A^{D}.}$