<s>
(	O
also	O
called	O
bistochastic	B-Algorithm
matrix	I-Algorithm
)	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
nonnegative	O
real	O
numbers	O
,	O
each	O
of	O
whose	O
rows	O
and	O
columns	O
sums	O
to	O
1	O
,	O
i.e.	O
,	O
</s>
<s>
Thus	O
,	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
both	O
left	O
stochastic	B-Algorithm
and	O
right	O
stochastic	B-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
any	O
matrix	O
that	O
is	O
both	O
left	O
and	O
right	O
stochastic	B-Algorithm
must	O
be	O
square	B-Algorithm
:	O
if	O
every	O
row	O
sums	O
to	O
1	O
then	O
the	O
sum	O
of	O
all	O
entries	O
in	O
the	O
matrix	O
must	O
be	O
equal	O
to	O
the	O
number	O
of	O
rows	O
,	O
and	O
since	O
the	O
same	O
holds	O
for	O
columns	O
,	O
the	O
number	O
of	O
rows	O
and	O
columns	O
must	O
be	O
equal	O
.	O
</s>
<s>
The	O
class	O
of	O
doubly	O
stochastic	B-Algorithm
matrices	I-Algorithm
is	O
a	O
convex	O
polytope	O
known	O
as	O
the	O
Birkhoff	O
polytope	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
–	I-Algorithm
von	I-Algorithm
Neumann	I-Algorithm
theorem	I-Algorithm
(	O
often	O
known	O
simply	O
as	O
Birkhoff	O
's	O
theorem	O
)	O
states	O
that	O
the	O
polytope	O
is	O
the	O
convex	O
hull	O
of	O
the	O
set	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
and	O
furthermore	O
that	O
the	O
vertices	O
of	O
are	O
precisely	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
two	O
doubly	O
stochastic	B-Algorithm
matrices	I-Algorithm
is	O
doubly	O
stochastic	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
inverse	O
of	O
a	O
nonsingular	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
need	O
not	O
be	O
doubly	O
stochastic	B-Algorithm
(	O
indeed	O
,	O
the	O
inverse	O
is	O
doubly	O
stochastic	B-Algorithm
iff	O
it	O
has	O
nonnegative	O
entries	O
)	O
.	O
</s>
<s>
The	O
stationary	O
distribution	O
of	O
an	O
irreducible	O
aperiodic	O
finite	O
Markov	O
chain	O
is	O
uniform	O
if	O
and	O
only	O
if	O
its	O
transition	O
matrix	O
is	O
doubly	O
stochastic	B-Algorithm
.	O
</s>
<s>
Sinkhorn	O
's	O
theorem	O
states	O
that	O
any	O
matrix	O
with	O
strictly	O
positive	O
entries	O
can	O
be	O
made	O
doubly	O
stochastic	B-Algorithm
by	O
pre	O
-	O
and	O
post-multiplication	O
by	O
diagonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
For	O
,	O
all	O
bistochastic	O
matrices	O
are	O
unistochastic	B-Algorithm
and	O
orthostochastic	B-Algorithm
,	O
but	O
for	O
larger	O
this	O
is	O
not	O
the	O
case	O
.	O
</s>
<s>
Van	O
der	O
Waerden	O
's	O
conjecture	O
that	O
the	O
minimum	O
permanent	O
among	O
all	O
doubly	O
stochastic	B-Algorithm
matrices	I-Algorithm
is	O
,	O
achieved	O
by	O
the	O
matrix	O
for	O
which	O
all	O
entries	O
are	O
equal	O
to	O
.	O
</s>
<s>
Let	O
X	O
be	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
we	O
will	O
show	O
that	O
there	O
exists	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
P	O
such	O
that	O
xij0	O
whenever	O
pij0	O
.	O
</s>
<s>
Thus	O
if	O
we	O
let	O
be	O
the	O
smallest	O
xij	O
corresponding	O
to	O
a	O
non-zero	O
pij	O
,	O
the	O
difference	O
X	O
–	O
P	O
will	O
be	O
a	O
scalar	O
multiple	O
of	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
and	O
will	O
have	O
at	O
least	O
one	O
more	O
zero	O
cell	O
than	O
X	O
.	O
</s>
<s>
Accordingly	O
we	O
may	O
successively	O
reduce	O
the	O
number	O
of	O
non-zero	O
cells	O
in	O
X	O
by	O
removing	O
scalar	O
multiples	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
until	O
we	O
arrive	O
at	O
the	O
zero	O
matrix	O
,	O
at	O
which	O
point	O
we	O
will	O
have	O
constructed	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
equal	O
to	O
the	O
original	O
X	O
.	O
</s>
<s>
For	O
every	O
i	O
in	O
A	O
,	O
the	O
sum	O
over	O
j	O
in	O
A	O
 '	O
of	O
xij	O
is	O
1	O
,	O
since	O
all	O
columns	O
j	O
for	O
which	O
xij0	O
are	O
included	O
in	O
A	O
 '	O
,	O
and	O
X	O
is	O
doubly	O
stochastic	B-Algorithm
;	O
hence	O
|A|	O
is	O
the	O
sum	O
over	O
all	O
i∈A	O
,	O
j∈A	O
 '	O
of	O
xij	O
.	O
</s>
<s>
These	O
edges	O
define	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
whose	O
non-zero	O
cells	O
correspond	O
to	O
non-zero	O
cells	O
in	O
X	O
.	O
</s>
<s>
The	O
proof	O
is	O
to	O
replace	O
the	O
i	O
th	O
row	O
of	O
the	O
original	O
matrix	O
by	O
ri	O
separate	O
rows	O
,	O
each	O
equal	O
to	O
the	O
original	O
row	O
divided	O
by	O
ri	O
;	O
to	O
apply	O
Birkhoff	O
's	O
theorem	O
to	O
the	O
resulting	O
square	B-Algorithm
matrix	I-Algorithm
;	O
and	O
at	O
the	O
end	O
to	O
additively	O
recombine	O
the	O
ri	O
rows	O
into	O
a	O
single	O
i	O
th	O
row	O
.	O
</s>
