<s>
In	O
mathematics	O
,	O
and	O
in	O
particular	O
,	O
algebra	O
,	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
(	O
or	O
,	O
g-inverse	O
)	O
of	O
an	O
element	O
x	O
is	O
an	O
element	O
y	O
that	O
has	O
some	O
properties	O
of	O
an	O
inverse	O
element	O
but	O
not	O
necessarily	O
all	O
of	O
them	O
.	O
</s>
<s>
The	O
purpose	O
of	O
constructing	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
to	O
obtain	O
a	O
matrix	B-Architecture
that	O
can	O
serve	O
as	O
an	O
inverse	O
in	O
some	O
sense	O
for	O
a	O
wider	O
class	O
of	O
matrices	O
than	O
invertible	O
matrices	O
.	O
</s>
<s>
Generalized	B-Algorithm
inverses	I-Algorithm
can	O
be	O
defined	O
in	O
any	O
mathematical	O
structure	O
that	O
involves	O
associative	O
multiplication	O
,	O
that	O
is	O
,	O
in	O
a	O
semigroup	O
.	O
</s>
<s>
This	O
article	O
describes	O
generalized	B-Algorithm
inverses	I-Algorithm
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
A	O
matrix	B-Architecture
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
a	O
matrix	B-Architecture
if	O
A	O
generalized	B-Algorithm
inverse	I-Algorithm
exists	O
for	O
an	O
arbitrary	O
matrix	B-Architecture
,	O
and	O
when	O
a	O
matrix	B-Architecture
has	O
a	O
regular	O
inverse	O
,	O
this	O
inverse	O
is	O
its	O
unique	O
generalized	B-Algorithm
inverse	I-Algorithm
.	O
</s>
<s>
where	O
is	O
an	O
matrix	B-Architecture
and	O
the	O
column	O
space	O
of	O
.	O
</s>
<s>
Hence	O
we	O
can	O
define	O
the	O
generalized	B-Algorithm
inverse	I-Algorithm
as	O
follows	O
:	O
Given	O
an	O
matrix	B-Architecture
,	O
an	O
matrix	B-Architecture
is	O
said	O
to	O
be	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
if	O
The	O
matrix	B-Architecture
has	O
been	O
termed	O
a	O
regular	O
inverse	O
of	O
by	O
some	O
authors	O
.	O
</s>
<s>
Important	O
types	O
of	O
generalized	B-Algorithm
inverse	I-Algorithm
include	O
:	O
</s>
<s>
Right	O
inverse	O
:	O
If	O
the	O
matrix	B-Architecture
has	O
dimensions	O
and	O
,	O
then	O
there	O
exists	O
an	O
matrix	B-Architecture
called	O
the	O
right	O
inverse	O
of	O
such	O
that	O
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Left	O
inverse	O
:	O
If	O
the	O
matrix	B-Architecture
has	O
dimensions	O
and	O
,	O
then	O
there	O
exists	O
an	O
matrix	B-Architecture
called	O
the	O
left	O
inverse	O
of	O
such	O
that	O
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Some	O
generalized	B-Algorithm
inverses	I-Algorithm
are	O
defined	O
and	O
classified	O
based	O
on	O
the	O
Penrose	O
conditions	O
:	O
</s>
<s>
If	O
satisfies	O
the	O
first	O
condition	O
,	O
then	O
it	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
If	O
it	O
satisfies	O
the	O
first	O
two	O
conditions	O
,	O
then	O
it	O
is	O
a	O
reflexive	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
If	O
it	O
satisfies	O
all	O
four	O
conditions	O
,	O
then	O
it	O
is	O
the	O
pseudoinverse	B-Algorithm
of	O
,	O
which	O
is	O
denoted	O
by	O
and	O
also	O
known	O
as	O
the	O
Moore	O
–	O
Penrose	O
inverse	O
,	O
after	O
the	O
pioneering	O
works	O
by	O
E	O
.	O
H	O
.	O
Moore	O
and	O
Roger	O
Penrose	O
.	O
</s>
<s>
When	O
is	O
non-singular	O
,	O
any	O
generalized	B-Algorithm
inverse	I-Algorithm
and	O
is	O
therefore	O
unique	O
.	O
</s>
<s>
For	O
a	O
singular	O
,	O
some	O
generalised	B-Algorithm
inverses	I-Algorithm
,	O
such	O
as	O
the	O
Drazin	B-Algorithm
inverse	I-Algorithm
and	O
the	O
Moore	O
–	O
Penrose	O
inverse	O
,	O
are	O
unique	O
,	O
while	O
others	O
are	O
not	O
necessarily	O
uniquely	O
defined	O
.	O
</s>
<s>
Hence	O
,	O
is	O
a	O
reflexive	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
has	O
no	O
left	O
inverse	O
.	O
</s>
<s>
The	O
element	O
b	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
an	O
element	O
a	O
if	O
and	O
only	O
if	O
,	O
in	O
any	O
semigroup	O
(	O
or	O
ring	O
,	O
since	O
the	O
multiplication	O
function	O
in	O
any	O
ring	O
is	O
a	O
semigroup	O
)	O
.	O
</s>
<s>
The	O
generalized	B-Algorithm
inverses	I-Algorithm
of	O
the	O
element	O
3	O
in	O
the	O
ring	O
are	O
3	O
,	O
7	O
,	O
and	O
11	O
,	O
since	O
in	O
the	O
ring	O
:	O
</s>
<s>
The	O
generalized	B-Algorithm
inverses	I-Algorithm
of	O
the	O
element	O
4	O
in	O
the	O
ring	O
are	O
1	O
,	O
4	O
,	O
7	O
,	O
and	O
10	O
,	O
since	O
in	O
the	O
ring	O
:	O
</s>
<s>
If	O
an	O
element	O
a	O
in	O
a	O
semigroup	O
(	O
or	O
ring	O
)	O
has	O
an	O
inverse	O
,	O
the	O
inverse	O
must	O
be	O
the	O
only	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
this	O
element	O
,	O
like	O
the	O
elements	O
1	O
,	O
5	O
,	O
7	O
,	O
and	O
11	O
in	O
the	O
ring	O
.	O
</s>
<s>
In	O
the	O
ring	O
,	O
any	O
element	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
0	O
,	O
however	O
,	O
2	O
has	O
no	O
generalized	B-Algorithm
inverse	I-Algorithm
,	O
since	O
there	O
is	O
no	O
b	O
in	O
such	O
that	O
.	O
</s>
<s>
A	O
right	O
inverse	O
of	O
a	O
non-square	B-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
,	O
provided	O
has	O
full	O
row	O
rank	O
.	O
</s>
<s>
A	O
left	O
inverse	O
of	O
a	O
non-square	B-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
,	O
provided	O
has	O
full	O
column	O
rank	O
.	O
</s>
<s>
If	O
for	O
any	O
non-singular	O
matrices	O
and	O
,	O
then	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
for	O
arbitrary	O
and	O
.	O
</s>
<s>
Then	O
,	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
if	O
and	O
only	O
if	O
.	O
</s>
<s>
Any	O
generalized	B-Algorithm
inverse	I-Algorithm
can	O
be	O
used	O
to	O
determine	O
whether	O
a	O
system	O
of	O
linear	O
equations	O
has	O
any	O
solutions	O
,	O
and	O
if	O
so	O
to	O
give	O
all	O
of	O
them	O
.	O
</s>
<s>
parametric	O
on	O
the	O
arbitrary	O
vector	O
,	O
where	O
is	O
any	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
If	O
A	O
has	O
full	O
column	O
rank	O
,	O
the	O
bracketed	O
expression	O
in	O
this	O
equation	O
is	O
the	O
zero	O
matrix	B-Architecture
and	O
so	O
the	O
solution	O
is	O
unique	O
.	O
</s>
<s>
The	O
generalized	B-Algorithm
inverses	I-Algorithm
of	O
matrices	O
can	O
be	O
characterized	O
as	O
follows	O
.	O
</s>
<s>
Conversely	O
,	O
any	O
choice	O
of	O
,	O
,	O
and	O
for	O
matrix	B-Architecture
of	O
this	O
form	O
is	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
of	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
pseudoinverse	B-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
In	O
practical	O
applications	O
it	O
is	O
necessary	O
to	O
identify	O
the	O
class	O
of	O
matrix	B-Architecture
transformations	O
that	O
must	O
be	O
preserved	O
by	O
a	O
generalized	B-Algorithm
inverse	I-Algorithm
.	O
</s>
<s>
The	O
Drazin	B-Algorithm
inverse	I-Algorithm
,	O
satisfies	O
the	O
following	O
definition	O
of	O
consistency	O
with	O
respect	O
to	O
similarity	O
transformations	O
involving	O
a	O
nonsingular	O
matrix	B-Architecture
S	O
:	O
</s>
