<s>
In	O
mathematics	O
,	O
a	O
unimodular	B-Algorithm
matrix	I-Algorithm
M	O
is	O
a	O
square	B-Algorithm
integer	B-Algorithm
matrix	I-Algorithm
having	O
determinant	O
+1	O
or	O
−1	O
.	O
</s>
<s>
Equivalently	O
,	O
it	O
is	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
that	O
is	O
invertible	O
over	O
the	O
integers	O
:	O
there	O
is	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
N	O
that	O
is	O
its	O
inverse	O
(	O
these	O
are	O
equivalent	O
under	O
Cramer	O
's	O
rule	O
)	O
.	O
</s>
<s>
Certain	O
transformation	B-Algorithm
matrices	I-Algorithm
for	O
rotation	B-Algorithm
,	O
shearing	B-Algorithm
(	O
both	O
with	O
determinant	O
1	O
)	O
and	O
reflection	B-Algorithm
(	O
determinant	O
−1	O
)	O
.	O
</s>
<s>
The	O
unimodular	B-Algorithm
matrix	I-Algorithm
used	O
(	O
possibly	O
implicitly	O
)	O
in	O
lattice	O
reduction	O
and	O
in	O
the	O
Hermite	O
normal	O
form	O
of	O
matrices	O
.	O
</s>
<s>
(	O
TU	O
matrix	O
)	O
is	O
a	O
matrix	O
for	O
which	O
every	O
square	B-Algorithm
non-singular	O
submatrix	O
is	O
unimodular	O
.	O
</s>
<s>
Equivalently	O
,	O
every	O
square	B-Algorithm
submatrix	O
has	O
determinant	O
0	O
,	O
+1	O
or	O
1	O
.	O
</s>
<s>
A	O
totally	B-Algorithm
unimodular	I-Algorithm
matrix	O
need	O
not	O
be	O
square	B-Algorithm
itself	O
.	O
</s>
<s>
From	O
the	O
definition	O
it	O
follows	O
that	O
any	O
submatrix	O
of	O
a	O
totally	B-Algorithm
unimodular	I-Algorithm
matrix	O
is	O
itself	O
totally	B-Algorithm
unimodular	I-Algorithm
(	O
TU	O
)	O
.	O
</s>
<s>
Totally	B-Algorithm
unimodular	I-Algorithm
matrices	O
are	O
extremely	O
important	O
in	O
polyhedral	O
combinatorics	O
and	O
combinatorial	O
optimization	O
since	O
they	O
give	O
a	O
quick	O
way	O
to	O
verify	O
that	O
a	O
linear	B-Algorithm
program	I-Algorithm
is	O
integral	O
(	O
has	O
an	O
integral	O
optimum	O
,	O
when	O
any	O
optimum	O
exists	O
)	O
.	O
</s>
<s>
The	O
unoriented	O
incidence	O
matrix	O
of	O
a	O
bipartite	O
graph	O
,	O
which	O
is	O
the	O
coefficient	O
matrix	O
for	O
bipartite	O
matching	O
,	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
(	O
TU	O
)	O
.	O
</s>
<s>
Let	O
be	O
an	O
m	O
by	O
n	O
matrix	O
whose	O
rows	O
can	O
be	O
partitioned	O
into	O
two	O
disjoint	B-Algorithm
sets	I-Algorithm
and	O
.	O
</s>
<s>
Then	O
the	O
following	O
four	O
conditions	O
together	O
are	O
sufficient	O
for	O
A	O
to	O
be	O
totally	B-Algorithm
unimodular	I-Algorithm
:	O
</s>
<s>
It	O
was	O
realized	O
later	O
that	O
these	O
conditions	O
define	O
an	O
incidence	O
matrix	O
of	O
a	O
balanced	O
signed	O
graph	O
;	O
thus	O
,	O
this	O
example	O
says	O
that	O
the	O
incidence	O
matrix	O
of	O
a	O
signed	O
graph	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
if	O
the	O
signed	O
graph	O
is	O
balanced	O
.	O
</s>
<s>
The	O
constraints	O
of	O
maximum	B-Algorithm
flow	I-Algorithm
and	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problems	I-Algorithm
yield	O
a	O
coefficient	O
matrix	O
with	O
these	O
properties	O
(	O
and	O
with	O
empty	O
C	O
)	O
.	O
</s>
<s>
Note	O
that	O
this	O
does	O
not	O
apply	O
to	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problems	I-Algorithm
,	O
in	O
which	O
it	O
is	O
possible	O
to	O
have	O
fractional	O
optimal	O
value	O
even	O
with	O
bounded	O
integer	O
capacities	O
.	O
</s>
<s>
Then	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
if	O
and	O
only	O
if	O
every	O
simple	O
arbitrarily-oriented	O
cycle	O
in	O
consists	O
of	O
alternating	O
forwards	O
and	O
backwards	O
arcs	O
.	O
</s>
<s>
The	O
following	O
matrix	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
:	O
</s>
<s>
This	O
matrix	O
arises	O
as	O
the	O
coefficient	O
matrix	O
of	O
the	O
constraints	O
in	O
the	O
linear	B-Algorithm
programming	I-Algorithm
formulation	I-Algorithm
of	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
on	O
the	O
following	O
network	O
:	O
</s>
<s>
is	O
not	O
totally	B-Algorithm
unimodular	I-Algorithm
,	O
since	O
it	O
has	O
a	O
square	B-Algorithm
submatrix	O
of	O
determinant−2	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
a	O
unimodular	B-Algorithm
matrix	I-Algorithm
is	O
one	O
that	O
is	O
invertible	O
over	O
the	O
ring	O
;	O
equivalently	O
,	O
whose	O
determinant	O
is	O
a	O
unit	O
.	O
</s>
<s>
A	O
rectangular	O
-by	O
-	O
matrix	O
is	O
said	O
to	O
be	O
unimodular	O
if	O
it	O
can	O
be	O
extended	O
with	O
rows	O
in	O
to	O
a	O
unimodular	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
