<s>
and	O
is	O
the	O
-by	O
-	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
condition	O
that	O
be	O
Hamiltonian	O
is	O
equivalent	O
to	O
requiring	O
that	O
the	O
matrices	O
and	O
are	O
symmetric	B-Algorithm
,	O
and	O
that	O
.	O
</s>
<s>
Another	O
equivalent	O
condition	O
is	O
that	O
is	O
of	O
the	O
form	O
with	O
symmetric	B-Algorithm
.	O
</s>
<s>
It	O
follows	O
easily	O
from	O
the	O
definition	O
that	O
the	O
transpose	O
of	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
is	O
Hamiltonian	O
.	O
</s>
<s>
This	O
group	O
consists	O
of	O
the	O
symplectic	B-Algorithm
matrices	I-Algorithm
,	O
those	O
matrices	O
which	O
satisfy	O
.	O
</s>
<s>
Thus	O
,	O
the	O
matrix	B-Architecture
exponential	O
of	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
is	O
symplectic	O
.	O
</s>
<s>
However	O
the	O
logarithm	O
of	O
a	O
symplectic	B-Algorithm
matrix	I-Algorithm
is	O
not	O
necessarily	O
Hamiltonian	O
because	O
the	O
exponential	O
map	O
from	O
the	O
Lie	O
algebra	O
to	O
the	O
group	O
is	O
not	O
surjective	O
.	O
</s>
<s>
The	O
characteristic	O
polynomial	O
of	O
a	O
real	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
is	O
even	O
.	O
</s>
<s>
Thus	O
,	O
if	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
has	O
as	O
an	O
eigenvalue	O
,	O
then	O
,	O
and	O
are	O
also	O
eigenvalues	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
trace	O
of	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
is	O
zero	O
.	O
</s>
<s>
The	O
square	O
of	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
is	O
skew-Hamiltonian	O
(	O
a	O
matrix	B-Architecture
is	O
skew-Hamiltonian	O
if	O
)	O
.	O
</s>
<s>
Conversely	O
,	O
every	O
skew-Hamiltonian	O
matrix	O
arises	O
as	O
the	O
square	O
of	O
a	O
Hamiltonian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
for	O
symplectic	B-Algorithm
matrices	I-Algorithm
,	O
the	O
definition	O
for	O
Hamiltonian	O
matrices	O
can	O
be	O
extended	O
to	O
complex	O
matrices	O
in	O
two	O
ways	O
.	O
</s>
<s>
One	O
possibility	O
is	O
to	O
say	O
that	O
a	O
matrix	B-Architecture
is	O
Hamiltonian	O
if	O
,	O
as	O
above	O
.	O
</s>
<s>
Another	O
possibility	O
is	O
to	O
use	O
the	O
condition	O
where	O
the	O
superscript	O
asterisk	O
(	O
)	O
denotes	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
A	O
linear	O
map	O
is	O
called	O
a	O
Hamiltonian	O
operator	O
with	O
respect	O
to	O
if	O
the	O
form	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
operator	O
is	O
Hamiltonian	O
with	O
respect	O
to	O
if	O
and	O
only	O
if	O
its	O
matrix	B-Architecture
in	O
this	O
basis	O
is	O
Hamiltonian	O
.	O
</s>
