<s>
In	O
mathematics	O
,	O
particularly	O
in	O
matrix	B-Architecture
theory	I-Architecture
,	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	O
binary	B-Algorithm
matrix	I-Algorithm
that	O
has	O
exactly	O
one	O
entry	O
of	O
1	O
in	O
each	O
row	O
and	O
each	O
column	O
and	O
0s	O
elsewhere	O
.	O
</s>
<s>
Each	O
such	O
matrix	O
,	O
say	O
,	O
represents	O
a	O
permutation	B-Algorithm
of	O
elements	O
and	O
,	O
when	O
used	O
to	O
multiply	O
another	O
matrix	O
,	O
say	O
,	O
results	O
in	O
permuting	O
the	O
rows	O
(	O
when	O
pre-multiplying	O
,	O
to	O
form	O
)	O
or	O
columns	O
(	O
when	O
post-multiplying	O
,	O
to	O
form	O
)	O
of	O
the	O
matrix	O
.	O
</s>
<s>
Given	O
a	O
permutation	B-Algorithm
of	O
m	O
elements	O
,	O
</s>
<s>
there	O
are	O
two	O
natural	O
ways	O
to	O
associate	O
the	O
permutation	B-Algorithm
with	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
;	O
namely	O
,	O
starting	O
with	O
the	O
m	O
×	O
m	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
,	O
either	O
permute	O
the	O
columns	O
or	O
permute	O
the	O
rows	O
,	O
according	O
to	O
.	O
</s>
<s>
Both	O
methods	O
of	O
defining	O
permutation	B-Algorithm
matrices	I-Algorithm
appear	O
in	O
the	O
literature	O
and	O
the	O
properties	O
expressed	O
in	O
one	O
representation	O
can	O
be	O
easily	O
converted	O
to	O
the	O
other	O
representation	O
.	O
</s>
<s>
The	O
m	O
×	O
m	O
permutation	B-Algorithm
matrix	I-Algorithm
P	O
=	O
(	O
pij	O
)	O
obtained	O
by	O
permuting	O
the	O
columns	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
that	O
is	O
,	O
for	O
each	O
i	O
,	O
if	O
j	O
=	O
(	O
i	O
)	O
and	O
otherwise	O
,	O
will	O
be	O
referred	O
to	O
as	O
the	O
column	O
representation	O
in	O
this	O
article	O
.	O
</s>
<s>
Observe	O
that	O
the	O
jth	O
column	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
now	O
appears	O
as	O
the	O
(	O
j	O
)	O
th	O
column	O
of	O
P	O
.	O
</s>
<s>
The	O
other	O
representation	O
,	O
obtained	O
by	O
permuting	O
the	O
rows	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
that	O
is	O
,	O
for	O
each	O
j	O
,	O
pij	O
=	O
1	O
if	O
i	O
=	O
(	O
j	O
)	O
and	O
otherwise	O
,	O
will	O
be	O
referred	O
to	O
as	O
the	O
row	O
representation	O
.	O
</s>
<s>
The	O
column	O
representation	O
of	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
used	O
throughout	O
this	O
section	O
,	O
except	O
when	O
otherwise	O
indicated	O
.	O
</s>
<s>
Repeated	O
use	O
of	O
this	O
result	O
shows	O
that	O
if	O
is	O
an	O
appropriately	O
sized	O
matrix	O
,	O
the	O
product	O
,	O
is	O
just	O
a	O
permutation	B-Algorithm
of	O
the	O
rows	O
of	O
.	O
</s>
<s>
for	O
each	O
shows	O
that	O
the	O
permutation	B-Algorithm
of	O
the	O
rows	O
is	O
given	O
by	O
−1	O
.	O
</s>
<s>
Again	O
,	O
repeated	O
application	O
of	O
this	O
result	O
shows	O
that	O
post-multiplying	O
a	O
matrix	O
by	O
the	O
permutation	B-Algorithm
matrix	I-Algorithm
,	O
that	O
is	O
,	O
,	O
results	O
in	O
permuting	O
the	O
columns	O
of	O
.	O
</s>
<s>
To	O
be	O
clear	O
,	O
the	O
above	O
formulas	O
use	O
the	O
prefix	O
notation	O
for	O
permutation	B-Algorithm
composition	O
,	O
that	O
is	O
,	O
</s>
<s>
Let	O
be	O
the	O
permutation	B-Algorithm
matrix	I-Algorithm
corresponding	O
to	O
in	O
its	O
row	O
representation	O
.	O
</s>
<s>
Permutation	B-Algorithm
matrices	I-Algorithm
can	O
be	O
characterized	O
as	O
the	O
orthogonal	B-Algorithm
matrices	I-Algorithm
whose	O
entries	O
are	O
all	O
non-negative	O
.	O
</s>
<s>
If	O
(	O
1	O
)	O
denotes	O
the	O
identity	O
permutation	B-Algorithm
,	O
then	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Let	O
denote	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
or	O
group	B-Algorithm
of	I-Algorithm
permutations	I-Algorithm
,	O
on	O
 { 1 , 2 , ... ,  } 	O
.	O
</s>
<s>
Since	O
there	O
are	O
permutations	B-Algorithm
,	O
there	O
are	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
By	O
the	O
formulas	O
above	O
,	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
form	O
a	O
group	O
under	O
matrix	O
multiplication	O
with	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
as	O
the	O
identity	O
element	O
.	O
</s>
<s>
The	O
map	O
that	O
sends	O
a	O
permutation	B-Algorithm
to	O
its	O
column	O
representation	O
is	O
a	O
faithful	O
representation	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
itself	O
a	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
,	O
but	O
it	O
also	O
plays	O
a	O
special	O
role	O
in	O
the	O
theory	O
of	O
these	O
matrices	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
–	I-Algorithm
von	I-Algorithm
Neumann	I-Algorithm
theorem	I-Algorithm
says	O
that	O
every	O
doubly	O
stochastic	O
real	O
matrix	O
is	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
of	O
the	O
same	O
order	O
and	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
the	O
extreme	O
points	O
of	O
the	O
set	O
of	O
doubly	O
stochastic	O
matrices	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
,	O
the	O
set	O
of	O
doubly	O
stochastic	O
matrices	O
,	O
is	O
the	O
convex	O
hull	O
of	O
the	O
set	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
trace	O
of	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
the	O
number	O
of	O
fixed	O
points	O
of	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
If	O
the	O
permutation	B-Algorithm
has	O
fixed	O
points	O
,	O
so	O
it	O
can	O
be	O
written	O
in	O
cycle	O
form	O
as	O
where	O
has	O
no	O
fixed	O
points	O
,	O
then	O
are	O
eigenvectors	O
of	O
the	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
To	O
calculate	O
the	O
eigenvalues	O
of	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
,	O
write	O
as	O
a	O
product	O
of	O
cycles	B-Algorithm
,	O
say	O
,	O
.	O
</s>
<s>
Let	O
the	O
corresponding	O
lengths	O
of	O
these	O
cycles	B-Algorithm
be	O
,	O
and	O
let	O
be	O
the	O
set	O
of	O
complex	O
solutions	O
of	O
.	O
</s>
<s>
The	O
union	O
of	O
all	O
s	O
is	O
the	O
set	O
of	O
eigenvalues	O
of	O
the	O
corresponding	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
From	O
group	O
theory	O
we	O
know	O
that	O
any	O
permutation	B-Algorithm
may	O
be	O
written	O
as	O
a	O
product	O
of	O
transpositions	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
any	O
permutation	B-Algorithm
matrix	I-Algorithm
factors	O
as	O
a	O
product	O
of	O
row-interchanging	O
elementary	O
matrices	O
,	O
each	O
having	O
determinant	O
1	O
.	O
</s>
<s>
Thus	O
,	O
the	O
determinant	O
of	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
the	O
signature	O
of	O
the	O
corresponding	O
permutation	B-Algorithm
.	O
</s>
<s>
When	O
a	O
matrix	O
M	O
is	O
multiplied	O
by	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
P	O
on	O
the	O
left	O
to	O
make	O
PM	O
,	O
the	O
product	O
is	O
the	O
result	O
of	O
permuting	O
the	O
rows	O
of	O
M	O
.	O
As	O
a	O
special	O
case	O
,	O
if	O
M	O
is	O
a	O
column	O
vector	O
,	O
then	O
PM	O
is	O
the	O
result	O
of	O
permuting	O
the	O
entries	O
of	O
M	O
:	O
</s>
<s>
When	O
instead	O
M	O
is	O
multiplied	O
by	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
on	O
the	O
right	O
to	O
make	O
MP	O
,	O
the	O
product	O
is	O
the	O
result	O
of	O
permuting	O
the	O
columns	O
of	O
M	O
.	O
As	O
a	O
special	O
case	O
,	O
if	O
M	O
is	O
a	O
row	O
vector	O
,	O
then	O
MP	O
is	O
the	O
result	O
of	O
permuting	O
the	O
entries	O
of	O
M	O
:	O
</s>
<s>
is	O
the	O
permutation	B-Algorithm
form	O
of	O
the	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
So	O
,	O
permutation	B-Algorithm
matrices	I-Algorithm
do	O
indeed	O
permute	O
the	O
order	O
of	O
elements	O
in	O
vectors	O
multiplied	O
with	O
them	O
.	O
</s>
