<s>
In	O
mathematics	O
and	O
theoretical	O
physics	O
,	O
a	O
supermatrix	B-Algorithm
is	O
a	O
Z2-graded	O
analog	O
of	O
an	O
ordinary	O
matrix	B-Architecture
.	O
</s>
<s>
Specifically	O
,	O
a	O
supermatrix	B-Algorithm
is	O
a	O
22	O
block	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
in	O
a	O
superalgebra	O
(	O
or	O
superring	O
)	O
.	O
</s>
<s>
Supermatrices	B-Algorithm
arise	O
in	O
the	O
study	O
of	O
super	O
linear	O
algebra	O
where	O
they	O
appear	O
as	O
the	O
coordinate	O
representations	O
of	O
a	O
linear	B-Architecture
transformations	I-Architecture
between	O
finite-dimensional	O
super	O
vector	O
spaces	O
or	O
free	O
supermodules	O
.	O
</s>
<s>
An	O
ordinary	O
(	O
ungraded	O
)	O
matrix	B-Architecture
can	O
be	O
thought	O
of	O
as	O
a	O
supermatrix	B-Algorithm
for	O
which	O
q	O
and	O
s	O
are	O
both	O
zero	O
.	O
</s>
<s>
A	O
square	B-Algorithm
supermatrix	B-Algorithm
is	O
one	O
for	O
which	O
(	O
r|s	O
)	O
=	O
(	O
p|q	O
)	O
.	O
</s>
<s>
This	O
means	O
that	O
not	O
only	O
is	O
the	O
unpartitioned	O
matrix	B-Architecture
X	O
square	B-Algorithm
,	O
but	O
the	O
diagonal	O
blocks	O
X00	O
and	O
X11	O
are	O
as	O
well	O
.	O
</s>
<s>
An	O
even	O
supermatrix	B-Algorithm
is	O
one	O
for	O
which	O
the	O
diagonal	O
blocks	O
(	O
X00	O
and	O
X11	O
)	O
consist	O
solely	O
of	O
even	O
elements	O
of	O
R	O
(	O
i.e.	O
</s>
<s>
An	O
odd	O
supermatrix	B-Algorithm
is	O
one	O
for	O
which	O
the	O
reverse	O
holds	O
:	O
the	O
diagonal	O
blocks	O
are	O
odd	O
and	O
the	O
off-diagonal	O
blocks	O
are	O
even	O
.	O
</s>
<s>
If	O
the	O
scalars	O
R	O
are	O
purely	O
even	O
there	O
are	O
no	O
nonzero	O
odd	O
elements	O
,	O
so	O
the	O
even	O
supermatices	O
are	O
the	O
block	B-Algorithm
diagonal	I-Algorithm
ones	O
and	O
the	O
odd	O
supermatrices	B-Algorithm
are	O
the	O
off-diagonal	O
ones	O
.	O
</s>
<s>
A	O
supermatrix	B-Algorithm
is	O
homogeneous	O
if	O
it	O
is	O
either	O
even	O
or	O
odd	O
.	O
</s>
<s>
The	O
parity	O
,	O
|X|	O
,	O
of	O
a	O
nonzero	O
homogeneous	O
supermatrix	B-Algorithm
X	O
is	O
0	O
or	O
1	O
according	O
to	O
whether	O
it	O
is	O
even	O
or	O
odd	O
.	O
</s>
<s>
Every	O
supermatrix	B-Algorithm
can	O
be	O
written	O
uniquely	O
as	O
the	O
sum	O
of	O
an	O
even	O
supermatrix	B-Algorithm
and	O
an	O
odd	O
one	O
.	O
</s>
<s>
Supermatrices	B-Algorithm
of	O
compatible	O
dimensions	O
can	O
be	O
added	O
or	O
multiplied	O
just	O
as	O
for	O
ordinary	O
matrices	O
.	O
</s>
<s>
One	O
can	O
also	O
multiply	O
supermatrices	B-Algorithm
by	O
elements	O
of	O
R	O
(	O
on	O
the	O
left	O
or	O
right	O
)	O
,	O
however	O
,	O
this	O
operation	O
differs	O
from	O
the	O
ungraded	O
case	O
due	O
to	O
the	O
presence	O
of	O
odd	O
elements	O
in	O
R	O
.	O
</s>
<s>
Let	O
Mr|s×p|	O
q(R )	O
denote	O
the	O
set	O
of	O
all	O
supermatrices	B-Algorithm
over	O
R	O
with	O
dimension	O
(	O
r|s	O
)	O
×( p|q	O
)	O
.	O
</s>
<s>
This	O
set	O
forms	O
a	O
supermodule	O
over	O
R	O
under	O
supermatrix	B-Algorithm
addition	O
and	O
scalar	O
multiplication	O
.	O
</s>
<s>
Let	O
Mp|	O
q(R )	O
denote	O
the	O
set	O
of	O
all	O
square	B-Algorithm
supermatices	O
over	O
R	O
with	O
dimension	O
(	O
p|q	O
)	O
×( p|q	O
)	O
.	O
</s>
<s>
This	O
set	O
forms	O
a	O
superring	O
under	O
supermatrix	B-Algorithm
addition	O
and	O
multiplication	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
R	O
is	O
a	O
commutative	O
superalgebra	O
,	O
then	O
supermatrix	B-Algorithm
multiplication	O
is	O
a	O
bilinear	O
operation	O
,	O
so	O
that	O
Mp|	O
q(R )	O
forms	O
a	O
superalgebra	O
over	O
R	O
.	O
</s>
<s>
Two	O
supermatrices	B-Algorithm
of	O
dimension	O
(	O
r|s	O
)	O
(	O
p|q	O
)	O
can	O
be	O
added	O
just	O
as	O
in	O
the	O
ungraded	O
case	O
to	O
obtain	O
a	O
supermatrix	B-Algorithm
of	O
the	O
same	O
dimension	O
.	O
</s>
<s>
It	O
is	O
easy	O
to	O
see	O
that	O
the	O
sum	O
of	O
two	O
even	O
supermatrices	B-Algorithm
is	O
even	O
and	O
the	O
sum	O
of	O
two	O
odd	O
supermatrices	B-Algorithm
is	O
odd	O
.	O
</s>
<s>
One	O
can	O
multiply	O
a	O
supermatrix	B-Algorithm
with	O
dimensions	O
(	O
r|s	O
)	O
(	O
p|q	O
)	O
by	O
a	O
supermatrix	B-Algorithm
with	O
dimensions	O
(	O
p|q	O
)	O
(	O
k|l	O
)	O
as	O
in	O
the	O
ungraded	O
case	O
to	O
obtain	O
a	O
matrix	B-Architecture
of	O
dimension	O
(	O
r|s	O
)	O
(	O
k|l	O
)	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
product	O
of	O
two	O
even	O
or	O
two	O
odd	O
supermatrices	B-Algorithm
is	O
even	O
while	O
the	O
product	O
of	O
an	O
even	O
and	O
odd	O
supermatrix	B-Algorithm
is	O
odd	O
.	O
</s>
<s>
Scalar	O
multiplication	O
for	O
supermatrices	B-Algorithm
is	O
different	O
than	O
the	O
ungraded	O
case	O
due	O
to	O
the	O
presence	O
of	O
odd	O
elements	O
in	O
R	O
.	O
Let	O
X	O
be	O
a	O
supermatrix	B-Algorithm
.	O
</s>
<s>
Ordinary	O
matrices	O
can	O
be	O
thought	O
of	O
as	O
the	O
coordinate	O
representations	O
of	O
linear	B-Architecture
maps	I-Architecture
between	O
vector	O
spaces	O
(	O
or	O
free	O
modules	O
)	O
.	O
</s>
<s>
Likewise	O
,	O
supermatrices	B-Algorithm
can	O
be	O
thought	O
of	O
as	O
the	O
coordinate	O
representations	O
of	O
linear	B-Architecture
maps	I-Architecture
between	O
super	O
vector	O
spaces	O
(	O
or	O
free	O
supermodules	O
)	O
.	O
</s>
<s>
The	O
coordinate	O
representation	O
of	O
such	O
a	O
transformation	O
is	O
always	O
an	O
even	O
supermatrix	B-Algorithm
.	O
</s>
<s>
Odd	O
supermatrices	B-Algorithm
correspond	O
to	O
linear	B-Architecture
transformations	I-Architecture
that	O
reverse	O
the	O
grading	O
.	O
</s>
<s>
General	O
supermatrices	B-Algorithm
represent	O
an	O
arbitrary	O
ungraded	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
where	O
the	O
action	O
of	O
T	O
on	O
Rp|q	O
is	O
just	O
supermatrix	B-Algorithm
multiplication	O
(	O
this	O
action	O
is	O
not	O
generally	O
left	O
R-linear	O
which	O
is	O
why	O
we	O
think	O
of	O
Rp|q	O
as	O
a	O
right	O
supermodule	O
)	O
.	O
</s>
<s>
can	O
be	O
written	O
as	O
a	O
(	O
r|s	O
)	O
(	O
p|q	O
)	O
supermatrix	B-Algorithm
relative	O
to	O
the	O
chosen	O
bases	O
.	O
</s>
<s>
The	O
block	O
decomposition	O
of	O
a	O
supermatrix	B-Algorithm
T	O
corresponds	O
to	O
the	O
decomposition	O
of	O
M	O
and	O
N	O
into	O
even	O
and	O
odd	O
submodules	O
:	O
</s>
<s>
Many	O
operations	O
on	O
ordinary	O
matrices	O
can	O
be	O
generalized	O
to	O
supermatrices	B-Algorithm
,	O
although	O
the	O
generalizations	O
are	O
not	O
always	O
obvious	O
or	O
straightforward	O
.	O
</s>
<s>
The	O
supertranspose	O
of	O
a	O
supermatrix	B-Algorithm
is	O
the	O
Z2-graded	O
analog	O
of	O
the	O
transpose	O
.	O
</s>
<s>
be	O
a	O
homogeneous	O
(	O
r|s	O
)	O
(	O
p|q	O
)	O
supermatrix	B-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
extended	O
to	O
arbitrary	O
supermatrices	B-Algorithm
by	O
linearity	O
.	O
</s>
<s>
Unlike	O
the	O
ordinary	O
transpose	O
,	O
the	O
supertranspose	O
is	O
not	O
generally	O
an	O
involution	B-Algorithm
,	O
but	O
rather	O
has	O
order	O
4	O
.	O
</s>
<s>
The	O
parity	O
transpose	O
of	O
a	O
supermatrix	B-Algorithm
is	O
a	O
new	O
operation	O
without	O
an	O
ungraded	O
analog	O
.	O
</s>
<s>
be	O
a	O
(	O
r|s	O
)	O
(	O
p|q	O
)	O
supermatrix	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
the	O
(	O
i	O
,	O
j	O
)	O
block	O
of	O
the	O
transposed	O
matrix	B-Architecture
is	O
the	O
(	O
1i	O
,	O
1j	O
)	O
block	O
of	O
the	O
original	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
supertrace	O
of	O
a	O
square	B-Algorithm
supermatrix	B-Algorithm
is	O
the	O
Z2-graded	O
analog	O
of	O
the	O
trace	O
.	O
</s>
<s>
for	O
homogeneous	O
supermatrices	B-Algorithm
X	O
and	O
Y	O
.	O
</s>
<s>
The	O
Berezinian	O
(	O
or	O
superdeterminant	O
)	O
of	O
a	O
square	B-Algorithm
supermatrix	B-Algorithm
is	O
the	O
Z2-graded	O
analog	O
of	O
the	O
determinant	O
.	O
</s>
<s>
where	O
det	O
denotes	O
the	O
ordinary	O
determinant	O
(	O
of	O
square	B-Algorithm
matrices	I-Algorithm
with	O
entries	O
in	O
the	O
commutative	O
algebra	O
R0	O
)	O
.	O
</s>
