<s>
In	O
mathematics	O
,	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
,	O
named	O
after	O
Michael	O
P	O
.	O
Drazin	O
,	O
is	O
a	O
kind	O
of	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Let	O
A	O
be	O
a	O
square	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
index	B-Architecture
of	O
A	O
is	O
the	O
least	O
nonnegative	O
integer	O
k	O
such	O
that	O
rank( Ak+1	O
)	O
=	O
rank(Ak )	O
.	O
</s>
<s>
It	O
's	O
not	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
in	O
the	O
classical	O
sense	O
,	O
since	O
in	O
general	O
.	O
</s>
<s>
If	O
is	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
of	O
,	O
then	O
is	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
The	O
Drazin	B-Algorithm
inverse	I-Algorithm
of	O
a	O
matrix	B-Architecture
of	O
index	B-Architecture
0	O
or	O
1	O
is	O
called	O
the	O
group	O
inverse	O
or	O
{1,2,5}-inverse	O
and	O
denoted	O
A#	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
matrix	I-Algorithm
P	O
,	O
defined	O
as	O
a	O
matrix	B-Architecture
such	O
that	O
P2	O
=	O
P	O
,	O
has	O
index	B-Architecture
1	O
(	O
or	O
0	O
)	O
and	O
has	O
Drazin	B-Algorithm
inverse	I-Algorithm
PD	O
=	O
P	O
.	O
</s>
<s>
For	O
or	O
any	O
regular	O
with	O
chosen	O
such	O
that	O
the	O
sequence	O
tends	O
to	O
its	O
Drazin	B-Algorithm
inverse	I-Algorithm
,	O
</s>
<s>
As	O
the	O
definition	O
of	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
is	O
invariant	O
under	O
matrix	B-Architecture
conjugations	O
,	O
writing	O
,	O
where	O
J	O
is	O
in	O
Jordan	O
normal	O
form	O
,	O
implies	O
that	O
.	O
</s>
<s>
The	O
Drazin	B-Algorithm
inverse	I-Algorithm
is	O
then	O
the	O
operation	O
that	O
maps	O
invertible	O
Jordan	O
blocks	O
to	O
their	O
inverses	O
,	O
and	O
nilpotent	O
Jordan	O
blocks	O
to	O
zero	O
.	O
</s>
<s>
More	O
generally	O
,	O
we	O
may	O
define	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
over	O
any	O
perfect	O
field	O
,	O
by	O
using	O
the	O
Jordan-Chevalley	O
decomposition	O
where	O
is	O
semisimple	O
and	O
is	O
nilpotent	O
and	O
both	O
operators	O
commute	O
.	O
</s>
<s>
The	O
Drazin	B-Algorithm
inverse	I-Algorithm
in	O
the	O
same	O
basis	O
is	O
then	O
defined	O
to	O
be	O
zero	O
on	O
the	O
kernel	O
of	O
,	O
and	O
equal	O
to	O
the	O
inverse	O
of	O
on	O
the	O
cokernel	O
of	O
.	O
</s>
