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

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


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


 * $$A = \begin{bmatrix}

B & 0  \\ 0  & N \end{bmatrix}$$ where $$B$$ is invertible with inverse $$B^{-1}$$ and $$N$$ is a nilpotent matrix, then


 * $$A^D = \begin{bmatrix}

B^{-1} & 0  \\ 0  & 0 \end{bmatrix}$$


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

The hyper-power sequence is
 * $$A_{i+1} := A_i + A_i\left(I - A A_i\right);$$ for convergence notice that $$A_{i+j} = A_i \sum_{k=0}^{2^j-1} \left(I - A A_i\right)^k.$$

For $$A_0 := \alpha A$$ or any regular $$A_0$$ with $$A_0 A = A A_0$$ chosen such that $$\left\|A_0 - A_0 A A_0\right\| < \left\|A_0\right\| $$ the sequence tends to its Drazin inverse,
 * $$A_i \rightarrow A^\text{D}.$$

Jordan normal form and Jordan-Chevalley decomposition
As the definition of the Drazin inverse is invariant under matrix conjugations, writing $$A = P J P^{-1}$$, where J is in Jordan normal form, implies that $$A^\text{D} = P J^\text{D} P^{-1} $$. The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.

More generally, we may define the Drazin inverse over any perfect field, by using the Jordan-Chevalley decomposition $$A =  A_s + A_n $$ where  $$A_s$$ is semisimple and $$A_n $$ is nilpotent and both operators commute. The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of $$A_s$$. The Drazin inverse in the same basis is then defined to be zero on the kernel of $$A_s$$, and equal to the inverse of $$A$$ on the cokernel of $$A_s$$.