<s>
In	O
mathematics	O
,	O
a	O
unistochastic	B-Algorithm
matrix	I-Algorithm
(	O
also	O
called	O
unitary-stochastic	O
)	O
is	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
whose	O
entries	O
are	O
the	O
squares	O
of	O
the	O
absolute	O
values	O
of	O
the	O
entries	O
of	O
some	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
definition	O
is	O
analogous	O
to	O
that	O
for	O
an	O
orthostochastic	B-Algorithm
matrix	I-Algorithm
,	O
which	O
is	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
whose	O
entries	O
are	O
the	O
squares	O
of	O
the	O
entries	O
in	O
some	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Since	O
all	O
orthogonal	B-Algorithm
matrices	I-Algorithm
are	O
necessarily	O
unitary	B-Algorithm
matrices	I-Algorithm
,	O
all	O
orthostochastic	B-Algorithm
matrices	O
are	O
also	O
unistochastic	B-Algorithm
.	O
</s>
<s>
First	O
,	O
all	O
2-by-2	O
doubly	O
stochastic	O
matrices	O
are	O
both	O
unistochastic	B-Algorithm
and	O
orthostochastic	B-Algorithm
,	O
but	O
for	O
larger	O
n	O
this	O
is	O
not	O
the	O
case	O
.	O
</s>
<s>
For	O
example	O
,	O
take	O
and	O
consider	O
the	O
following	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
This	O
matrix	O
is	O
not	O
unistochastic	B-Algorithm
,	O
since	O
any	O
two	O
vectors	O
with	O
moduli	O
equal	O
to	O
the	O
square	O
root	O
of	O
the	O
entries	O
of	O
two	O
columns	O
(	O
or	O
rows	O
)	O
of	O
B	O
cannot	O
be	O
made	O
orthogonal	O
by	O
a	O
suitable	O
choice	O
of	O
phases	O
.	O
</s>
<s>
For	O
,	O
the	O
set	O
of	O
orthostochastic	B-Algorithm
matrices	O
is	O
a	O
proper	O
subset	O
of	O
the	O
set	O
of	O
unistochastic	B-Algorithm
matrices	O
.	O
</s>
<s>
The	O
Schur-Horn	O
theorem	O
is	O
equivalent	O
to	O
the	O
following	O
"	O
weak	O
convexity	O
"	O
property	O
of	O
the	O
set	O
of	O
unistochastic	B-Algorithm
matrices	O
:	O
for	O
any	O
vector	O
the	O
set	O
is	O
the	O
convex	O
hull	O
of	O
the	O
set	O
of	O
vectors	O
obtained	O
by	O
all	O
permutations	O
of	O
the	O
entries	O
of	O
the	O
vector	O
(	O
the	O
permutation	O
polytope	O
generated	O
by	O
the	O
vector	O
)	O
.	O
</s>
<s>
The	O
set	O
of	O
unistochastic	B-Algorithm
matrices	O
has	O
a	O
nonempty	O
interior	O
.	O
</s>
<s>
The	O
unistochastic	B-Algorithm
matrix	I-Algorithm
corresponding	O
to	O
the	O
unitary	O
matrix	O
with	O
the	O
entries	O
,	O
where	O
and	O
,	O
is	O
an	O
interior	O
point	O
of	O
.	O
</s>
