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
.
