<s>
A	O
logical	B-Algorithm
matrix	I-Algorithm
,	O
binary	B-Algorithm
matrix	I-Algorithm
,	O
relation	O
matrix	B-Architecture
,	O
Boolean	O
matrix	B-Architecture
,	O
or	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
is	O
a	O
matrix	B-Architecture
with	O
entries	O
from	O
the	O
Boolean	O
domain	O
Such	O
a	O
matrix	B-Architecture
can	O
be	O
used	O
to	O
represent	O
a	O
binary	O
relation	O
between	O
a	O
pair	O
of	O
finite	O
sets	O
.	O
</s>
<s>
In	O
order	O
to	O
designate	O
the	O
row	O
and	O
column	O
numbers	O
of	O
the	O
matrix	B-Architecture
,	O
the	O
sets	O
X	O
and	O
Y	O
are	O
indexed	O
with	O
positive	O
integers	O
:	O
i	O
ranges	O
from	O
1	O
to	O
the	O
cardinality	B-Application
(	O
size	O
)	O
of	O
X	O
,	O
and	O
j	O
ranges	O
from	O
1	O
to	O
the	O
cardinality	B-Application
of	O
Y	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
,	O
all	O
of	O
whose	O
columns	O
and	O
rows	O
each	O
have	O
exactly	O
one	O
nonzero	O
element	O
.	O
</s>
<s>
A	O
Costas	B-Algorithm
array	I-Algorithm
is	O
a	O
special	O
case	O
of	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
incidence	B-Algorithm
matrix	I-Algorithm
in	O
combinatorics	O
and	O
finite	O
geometry	O
has	O
ones	O
to	O
indicate	O
incidence	O
between	O
points	O
(	O
or	O
vertices	O
)	O
and	O
lines	O
of	O
a	O
geometry	O
,	O
blocks	O
of	O
a	O
block	O
design	O
,	O
or	O
edges	O
of	O
a	O
graph	O
.	O
</s>
<s>
A	O
design	B-Algorithm
matrix	I-Algorithm
in	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
is	O
a	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
with	O
constant	O
row	O
sums	O
.	O
</s>
<s>
A	O
logical	B-Algorithm
matrix	I-Algorithm
may	O
represent	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
in	O
graph	O
theory	O
:	O
non-symmetric	O
matrices	O
correspond	O
to	O
directed	O
graphs	O
,	O
symmetric	B-Algorithm
matrices	I-Algorithm
to	O
ordinary	O
graphs	O
,	O
and	O
a	O
1	O
on	O
the	O
diagonal	O
corresponds	O
to	O
a	O
loop	O
at	O
the	O
corresponding	O
vertex	O
.	O
</s>
<s>
The	O
biadjacency	O
matrix	B-Architecture
of	O
a	O
simple	O
,	O
undirected	O
bipartite	O
graph	O
is	O
a	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
,	O
and	O
any	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
arises	O
in	O
this	O
way	O
.	O
</s>
<s>
The	O
prime	O
factors	O
of	O
a	O
list	O
of	O
m	O
square-free	O
,	O
n-smooth	O
numbers	O
can	O
be	O
described	O
as	O
an	O
m×	O
π(n )	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
,	O
where	O
π	O
is	O
the	O
prime-counting	O
function	O
,	O
and	O
aij	O
is	O
1	O
if	O
and	O
only	O
if	O
the	O
j	O
th	O
prime	O
divides	O
the	O
i	O
th	O
number	O
.	O
</s>
<s>
This	O
representation	O
is	O
useful	O
in	O
the	O
quadratic	B-Algorithm
sieve	I-Algorithm
factoring	O
algorithm	O
.	O
</s>
<s>
A	O
bitmap	B-Algorithm
image	I-Algorithm
containing	O
pixels	B-Algorithm
in	O
only	O
two	O
colors	O
can	O
be	O
represented	O
as	O
a	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
in	O
which	O
the	O
zeros	O
represent	O
pixels	B-Algorithm
of	O
one	O
color	O
and	O
the	O
ones	O
represent	O
pixels	B-Algorithm
of	O
the	O
other	O
color	O
.	O
</s>
<s>
A	O
binary	B-Algorithm
matrix	I-Algorithm
can	O
be	O
used	O
to	O
check	O
the	O
game	O
rules	O
in	O
the	O
game	O
of	O
Go	O
.	O
</s>
<s>
The	O
four	O
valued	O
logic	O
of	O
two	O
bits	O
,	O
transformed	O
by	O
2x2	O
logical	O
matrices	O
,	O
forms	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
.	O
</s>
<s>
The	O
matrix	B-Architecture
representation	O
of	O
the	O
equality	O
relation	O
on	O
a	O
finite	O
set	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
I	O
,	O
that	O
is	O
,	O
the	O
matrix	B-Architecture
whose	O
entries	O
on	O
the	O
diagonal	O
are	O
all	O
1	O
,	O
while	O
the	O
others	O
are	O
all	O
0	O
.	O
</s>
<s>
If	O
the	O
Boolean	O
domain	O
is	O
viewed	O
as	O
a	O
semiring	O
,	O
where	O
addition	O
corresponds	O
to	O
logical	O
OR	O
and	O
multiplication	O
to	O
logical	O
AND	O
,	O
the	O
matrix	B-Architecture
representation	O
of	O
the	O
composition	O
of	O
two	O
relations	O
is	O
equal	O
to	O
the	O
matrix	B-Architecture
product	O
of	O
the	O
matrix	B-Architecture
representations	O
of	O
these	O
relations	O
.	O
</s>
<s>
The	O
complement	O
of	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
is	O
obtained	O
by	O
swapping	O
all	O
zeros	O
and	O
ones	O
for	O
their	O
opposite	O
.	O
</s>
<s>
Every	O
logical	B-Algorithm
matrix	I-Algorithm
has	O
a	O
transpose	O
Suppose	O
A	O
is	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
with	O
no	O
columns	O
or	O
rows	O
identically	O
zero	O
.	O
</s>
<s>
Then	O
the	O
matrix	B-Architecture
product	O
,	O
using	O
Boolean	O
arithmetic	O
,	O
contains	O
the	O
m×m	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
the	O
product	O
contains	O
the	O
n×n	O
identity	O
.	O
</s>
<s>
As	O
a	O
mathematical	O
structure	O
,	O
the	O
Boolean	O
algebra	O
U	O
forms	O
a	O
lattice	O
ordered	O
by	O
inclusion	O
;	O
additionally	O
it	O
is	O
a	O
multiplicative	O
lattice	O
due	O
to	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
Every	O
logical	B-Algorithm
matrix	I-Algorithm
in	O
U	O
corresponds	O
to	O
a	O
binary	O
relation	O
.	O
</s>
<s>
These	O
listed	O
operations	O
on	O
U	O
,	O
and	O
ordering	O
,	O
correspond	O
to	O
a	O
calculus	O
of	O
relations	O
,	O
where	O
the	O
matrix	B-Architecture
multiplication	O
represents	O
composition	O
of	O
relations	O
.	O
</s>
<s>
If	O
m	O
or	O
n	O
equals	O
one	O
,	O
then	O
the	O
m×n	O
logical	B-Algorithm
matrix	I-Algorithm
(	O
mij	O
)	O
is	O
a	O
logical	O
vector	O
or	O
bit	B-Data_Structure
string	I-Data_Structure
.	O
</s>
<s>
A	O
reordering	O
of	O
the	O
rows	O
and	O
columns	O
of	O
such	O
a	O
matrix	B-Architecture
can	O
assemble	O
all	O
the	O
ones	O
into	O
a	O
rectangular	O
part	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
Consider	O
the	O
table	O
of	O
group-like	O
structures	O
,	O
where	O
"	O
unneeded	O
"	O
can	O
be	O
denoted	O
0	O
,	O
and	O
"	O
required	O
"	O
denoted	O
by	O
1	O
,	O
forming	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
To	O
calculate	O
elements	O
of	O
,	O
it	O
is	O
necessary	O
to	O
use	O
the	O
logical	O
inner	O
product	O
of	O
pairs	O
of	O
logical	O
vectors	O
in	O
rows	O
of	O
this	O
matrix	B-Architecture
.	O
</s>
<s>
In	O
fact	O
,	O
small	O
category	O
is	O
orthogonal	O
to	O
quasigroup	O
,	O
and	O
groupoid	O
is	O
orthogonal	O
to	O
magma	B-General_Concept
.	O
</s>
<s>
Adding	O
up	O
all	O
the	O
ones	O
in	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
may	O
be	O
accomplished	O
in	O
two	O
ways	O
:	O
first	O
summing	O
the	O
rows	O
or	O
first	O
summing	O
the	O
columns	O
.	O
</s>
<s>
In	O
incidence	O
geometry	O
,	O
the	O
matrix	B-Architecture
is	O
interpreted	O
as	O
an	O
incidence	B-Algorithm
matrix	I-Algorithm
with	O
the	O
rows	O
corresponding	O
to	O
"	O
points	O
"	O
and	O
the	O
columns	O
as	O
"	O
blocks	O
"	O
(	O
generalizing	O
lines	O
made	O
of	O
points	O
)	O
.	O
</s>
<s>
An	O
early	O
problem	O
in	O
the	O
area	O
was	O
"	O
to	O
find	O
necessary	O
and	O
sufficient	O
conditions	O
for	O
the	O
existence	O
of	O
an	O
incidence	O
structure	O
with	O
given	O
point	O
degrees	O
and	O
block	O
degrees	O
;	O
or	O
in	O
matrix	B-Architecture
language	O
,	O
for	O
the	O
existence	O
of	O
a	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
of	O
type	O
v×b	O
with	O
given	O
row	O
and	O
column	O
sums	O
"	O
.	O
</s>
