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(A^k+1) = rank(A^k). The Drazin inverse of A is the unique matrix A^D that satisfies

${\displaystyle A^{k+1}A^{\text{D}}=A^{k},\quad A^{\text{D}}AA^{\text{D}}=A^{\text{D}},\quad AA^{\text{D}}=A^{\text{D}}A.}$

It's not a generalized inverse in the classical sense, since ${\displaystyle AA^{\text{D}}A\neq A}$ in general.

• If A is invertible with inverse ${\displaystyle A^{-1}}$, then ${\displaystyle A^{\text{D}}=A^{-1}}$.
• If A is a block diagonal matrix

${\displaystyle A={\begin{bmatrix}B&0\\0&N\end{bmatrix}}}$

where ${\displaystyle B}$ is invertible with inverse ${\displaystyle B^{-1}}$ and ${\displaystyle N}$ is a nilpotent matrix, then

${\displaystyle A^{D}={\begin{bmatrix}B^{-1}&0\\0&0\end{bmatrix}}}$

• Drazin inversion is invariant under conjugation. If ${\displaystyle A^{\text{D}}}$ is the Drazin inverse of ${\displaystyle A}$, then ${\displaystyle PA^{\text{D}}P^{-1}}$ is the Drazin inverse of ${\displaystyle PAP^{-1}}$.
• 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 P^D = P.
• If A is a nilpotent matrix (for example a shift matrix), then ${\displaystyle A^{\text{D}}=0.}$

The hyper-power sequence is

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

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

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