<s>
where	O
denotes	O
the	O
transpose	O
of	O
and	O
is	O
a	O
fixed	O
nonsingular	O
,	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
matrix	B-Architecture
has	O
determinant	O
and	O
its	O
inverse	O
is	O
.	O
</s>
<s>
Every	O
symplectic	B-Algorithm
matrix	I-Algorithm
has	O
determinant	O
,	O
and	O
the	O
symplectic	O
matrices	O
with	O
real	O
entries	O
form	O
a	O
subgroup	O
of	O
the	O
general	O
linear	O
group	O
under	O
matrix	B-Architecture
multiplication	O
since	O
being	O
symplectic	O
is	O
a	O
property	O
stable	O
under	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
Topologically	B-Architecture
,	O
this	O
symplectic	O
group	O
is	O
a	O
connected	O
noncompact	O
real	O
Lie	O
group	O
of	O
real	O
dimension	O
,	O
and	O
is	O
denoted	O
.	O
</s>
<s>
The	O
symplectic	O
group	O
can	O
be	O
defined	O
as	O
the	O
set	O
of	O
linear	B-Architecture
transformations	I-Architecture
that	O
preserve	O
the	O
symplectic	O
form	O
of	O
a	O
real	O
symplectic	O
vector	O
space	O
.	O
</s>
<s>
where	O
is	O
the	O
set	O
of	O
symmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
any	O
symplectic	B-Algorithm
matrix	I-Algorithm
can	O
be	O
constructed	O
by	O
multiplying	O
matrices	O
in	O
and	O
together	O
,	O
along	O
with	O
some	O
power	O
of	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
product	O
of	O
two	O
symplectic	O
matrices	O
is	O
,	O
again	O
,	O
a	O
symplectic	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
natural	O
manifold	B-Architecture
structure	O
on	O
this	O
group	O
which	O
makes	O
it	O
into	O
a	O
(	O
real	O
or	O
complex	O
)	O
Lie	O
group	O
called	O
the	O
symplectic	O
group	O
.	O
</s>
<s>
It	O
follows	O
easily	O
from	O
the	O
definition	O
that	O
the	O
determinant	O
of	O
any	O
symplectic	B-Algorithm
matrix	I-Algorithm
is	O
±1	O
.	O
</s>
<s>
The	O
condition	O
for	O
to	O
be	O
symplectic	O
is	O
equivalent	O
to	O
the	O
two	O
following	O
equivalent	O
conditions	O
symmetric	B-Algorithm
,	O
and	O
symmetric	B-Algorithm
,	O
and	O
When	O
these	O
conditions	O
reduce	O
to	O
the	O
single	O
condition	O
.	O
</s>
<s>
Thus	O
a	O
matrix	B-Architecture
is	O
symplectic	O
iff	O
it	O
has	O
unit	O
determinant	O
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
by	O
noting	O
that	O
is	O
anti-symmetric	O
.	O
</s>
<s>
In	O
the	O
abstract	O
formulation	O
of	O
linear	B-Language
algebra	I-Language
,	O
matrices	O
are	O
replaced	O
with	O
linear	B-Architecture
transformations	I-Architecture
of	O
finite-dimensional	O
vector	O
spaces	O
.	O
</s>
<s>
The	O
abstract	O
analog	O
of	O
a	O
symplectic	B-Algorithm
matrix	I-Algorithm
is	O
a	O
symplectic	O
transformation	O
of	O
a	O
symplectic	O
vector	O
space	O
.	O
</s>
<s>
Briefly	O
,	O
a	O
symplectic	O
vector	O
space	O
is	O
a	O
-dimensional	O
vector	O
space	O
equipped	O
with	O
a	O
nondegenerate	O
,	O
skew-symmetric	B-Algorithm
bilinear	O
form	O
called	O
the	O
symplectic	O
form	O
.	O
</s>
<s>
A	O
symplectic	O
transformation	O
is	O
then	O
a	O
linear	B-Architecture
transformation	I-Architecture
which	O
preserves	O
,	O
i.e.	O
</s>
<s>
Fixing	O
a	O
basis	O
for	O
,	O
can	O
be	O
written	O
as	O
a	O
matrix	B-Architecture
and	O
as	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
condition	O
that	O
be	O
a	O
symplectic	O
transformation	O
is	O
precisely	O
the	O
condition	O
that	O
M	O
be	O
a	O
symplectic	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
One	O
can	O
always	O
bring	O
to	O
either	O
the	O
standard	O
form	O
given	O
in	O
the	O
introduction	O
or	O
the	O
block	B-Algorithm
diagonal	I-Algorithm
form	O
described	O
below	O
by	O
a	O
suitable	O
choice	O
of	O
A	O
.	O
</s>
<s>
Symplectic	O
matrices	O
are	O
defined	O
relative	O
to	O
a	O
fixed	O
nonsingular	O
,	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
explained	O
in	O
the	O
previous	O
section	O
,	O
can	O
be	O
thought	O
of	O
as	O
the	O
coordinate	O
representation	O
of	O
a	O
nondegenerate	O
skew-symmetric	B-Algorithm
bilinear	O
form	O
.	O
</s>
<s>
It	O
is	O
a	O
basic	O
result	O
in	O
linear	B-Language
algebra	I-Language
that	O
any	O
two	O
such	O
matrices	O
differ	O
from	O
each	O
other	O
by	O
a	O
change	O
of	O
basis	O
.	O
</s>
<s>
This	O
choice	O
differs	O
from	O
the	O
previous	O
one	O
by	O
a	O
permutation	B-Algorithm
of	O
basis	O
vectors	O
.	O
</s>
<s>
Sometimes	O
the	O
notation	O
is	O
used	O
instead	O
of	O
for	O
the	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
complex	O
structure	O
is	O
the	O
coordinate	O
representation	O
of	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
squares	O
to	O
,	O
whereas	O
is	O
the	O
coordinate	O
representation	O
of	O
a	O
nondegenerate	O
skew-symmetric	B-Algorithm
bilinear	O
form	O
.	O
</s>
<s>
One	O
could	O
easily	O
choose	O
bases	O
in	O
which	O
is	O
not	O
skew-symmetric	B-Algorithm
or	O
does	O
not	O
square	O
to	O
.	O
</s>
<s>
That	O
and	O
usually	O
have	O
the	O
same	O
coordinate	O
expression	O
(	O
up	O
to	O
an	O
overall	O
sign	O
)	O
is	O
simply	O
a	O
consequence	O
of	O
the	O
fact	O
that	O
the	O
metric	O
g	O
is	O
usually	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
the	O
diagonal	B-Algorithm
elements	O
of	O
are	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
Any	O
real	O
symplectic	B-Algorithm
matrix	I-Algorithm
has	O
a	O
polar	O
decomposition	O
of	O
the	O
form	O
:	O
</s>
<s>
Any	O
real	O
symplectic	B-Algorithm
matrix	I-Algorithm
can	O
be	O
decomposed	O
as	O
a	O
product	O
of	O
three	O
matrices	O
:	O
</s>
<s>
such	O
that	O
and	O
are	O
both	O
symplectic	O
and	O
orthogonal	B-Algorithm
and	O
is	O
positive-definite	B-Algorithm
and	O
diagonal	B-Algorithm
.	O
</s>
<s>
This	O
decomposition	O
is	O
closely	O
related	O
to	O
the	O
singular	O
value	O
decomposition	O
of	O
a	O
matrix	B-Architecture
and	O
is	O
known	O
as	O
an	O
'	O
Euler	O
 '	O
or	O
'	O
Bloch-Messiah	O
'	O
decomposition	O
.	O
</s>
<s>
If	O
instead	O
M	O
is	O
a	O
matrix	B-Architecture
with	O
complex	O
entries	O
,	O
the	O
definition	O
is	O
not	O
standard	O
throughout	O
the	O
literature	O
.	O
</s>
<s>
where	O
M*	O
denotes	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
M	O
.	O
In	O
this	O
case	O
,	O
the	O
determinant	O
may	O
not	O
be	O
1	O
,	O
but	O
will	O
have	O
absolute	O
value	O
1	O
.	O
</s>
<s>
In	O
the	O
2×2	O
case	O
(	O
n	O
=	O
1	O
)	O
,	O
M	O
will	O
be	O
the	O
product	O
of	O
a	O
real	O
symplectic	B-Algorithm
matrix	I-Algorithm
and	O
a	O
complex	O
number	O
of	O
absolute	O
value	O
1	O
.	O
</s>
<s>
In	O
turn	O
,	O
the	O
Bloch-Messiah	O
decomposition	O
(	O
)	O
means	O
that	O
such	O
an	O
arbitrary	O
Gaussian	O
transformation	O
can	O
be	O
represented	O
as	O
a	O
set	O
of	O
two	O
passive	O
linear-optical	O
interferometers	O
(	O
corresponding	O
to	O
orthogonal	B-Algorithm
matrices	I-Algorithm
O	O
and	O
O	O
 '	O
)	O
intermitted	O
by	O
a	O
layer	O
of	O
active	O
non-linear	O
squeezing	O
transformations	O
(	O
given	O
in	O
terms	O
of	O
the	O
matrix	B-Architecture
D	O
)	O
.	O
</s>
