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
A k + 1 A D = A k , A D A A D = A D , A A D = A D A . If A is invertible with inverse A − 1 , then 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 A D = 0. The hyper-power sequence is
A i + 1 := A i + A i ( I − A A i ) ; for convergence notice that
A i + j = A i ∑ k = 0 2 j − 1 ( I − A A i ) k . For A 0 := α A or any regular A 0 with A 0 A = A A 0 chosen such that ∥ A 0 − A 0 A A 0 ∥ < ∥ A 0 ∥ the sequence tends to its Drazin inverse,
A i → A D . 