<s>
In	O
mathematics	O
,	O
particularly	O
in	O
linear	B-Language
algebra	I-Language
,	O
a	O
skew-symmetric	O
(	O
or	O
antisymmetric	O
or	O
antimetric	O
)	O
matrix	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
whose	O
transpose	O
equals	O
its	O
negative	O
.	O
</s>
<s>
If	O
the	O
characteristic	O
of	O
the	O
field	O
is	O
2	O
,	O
then	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
the	O
same	O
thing	O
as	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
sum	O
of	O
two	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
is	O
skew-symmetric	O
.	O
</s>
<s>
A	O
scalar	O
multiple	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
skew-symmetric	O
.	O
</s>
<s>
The	O
elements	O
on	O
the	O
diagonal	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
are	O
zero	O
,	O
and	O
therefore	O
its	O
trace	O
equals	O
zero	O
.	O
</s>
<s>
If	O
is	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
and	O
is	O
a	O
real	O
eigenvalue	O
,	O
then	O
,	O
i.e.	O
</s>
<s>
the	O
nonzero	O
eigenvalues	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
are	O
non-real	O
.	O
</s>
<s>
If	O
is	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
,	O
then	O
is	O
invertible	O
,	O
where	O
is	O
the	O
identity	O
matrix	O
.	O
</s>
<s>
If	O
is	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
then	O
is	O
a	O
symmetric	B-Algorithm
negative	B-Algorithm
semi-definite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
first	O
two	O
properties	O
above	O
,	O
the	O
set	O
of	O
all	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
of	O
a	O
fixed	O
size	O
forms	O
a	O
vector	O
space	O
.	O
</s>
<s>
A	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
determined	O
by	O
scalars	O
(	O
the	O
number	O
of	O
entries	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
;	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
determined	O
by	O
scalars	O
(	O
the	O
number	O
of	O
entries	O
on	O
or	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
.	O
</s>
<s>
Let	O
denote	O
the	O
space	O
of	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
and	O
denote	O
the	O
space	O
of	O
symmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Notice	O
that	O
and	O
This	O
is	O
true	O
for	O
every	O
square	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
from	O
any	O
field	O
whose	O
characteristic	O
is	O
different	O
from	O
2	O
.	O
</s>
<s>
Since	O
this	O
definition	O
is	O
independent	O
of	O
the	O
choice	O
of	O
basis	O
,	O
skew-symmetry	B-Algorithm
is	O
a	O
property	O
that	O
depends	O
only	O
on	O
the	O
linear	B-Architecture
operator	I-Architecture
and	O
a	O
choice	O
of	O
inner	O
product	O
.	O
</s>
<s>
skew	B-Algorithm
symmetric	I-Algorithm
matrices	O
can	O
be	O
used	O
to	O
represent	O
cross	O
products	O
as	O
matrix	O
multiplications	O
.	O
</s>
<s>
Let	O
be	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Hence	O
,	O
all	O
odd	O
dimension	O
skew	B-Algorithm
symmetric	I-Algorithm
matrices	O
are	O
singular	O
as	O
their	O
determinants	O
are	O
always	O
zero	O
.	O
</s>
<s>
Thus	O
the	O
determinant	O
of	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
always	O
non-negative	O
.	O
</s>
<s>
However	O
this	O
last	O
fact	O
can	O
be	O
proved	O
in	O
an	O
elementary	O
way	O
as	O
follows	O
:	O
the	O
eigenvalues	O
of	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
are	O
purely	O
imaginary	O
(	O
see	O
below	O
)	O
and	O
to	O
every	O
eigenvalue	O
there	O
corresponds	O
the	O
conjugate	O
eigenvalue	O
with	O
the	O
same	O
multiplicity	O
;	O
therefore	O
,	O
as	O
the	O
determinant	O
is	O
the	O
product	O
of	O
the	O
eigenvalues	O
,	O
each	O
one	O
repeated	O
according	O
to	O
its	O
multiplicity	O
,	O
it	O
follows	O
at	O
once	O
that	O
the	O
determinant	O
,	O
if	O
it	O
is	O
not	O
0	O
,	O
is	O
a	O
positive	O
real	O
number	O
.	O
</s>
<s>
The	O
number	O
of	O
distinct	O
terms	O
in	O
the	O
expansion	O
of	O
the	O
determinant	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
of	O
order	O
has	O
been	O
considered	O
already	O
by	O
Cayley	O
,	O
Sylvester	O
,	O
and	O
Pfaff	O
.	O
</s>
<s>
Three-by-three	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
can	O
be	O
used	O
to	O
represent	O
cross	O
products	O
as	O
matrix	O
multiplications	O
.	O
</s>
<s>
i.e.	O
,	O
the	O
commutator	O
of	O
skew-symmetric	O
three-by-three	O
matrices	O
can	O
be	O
identified	O
with	O
the	O
cross-product	O
of	O
three-vectors	O
.	O
</s>
<s>
Since	O
the	O
skew-symmetric	O
three-by-three	O
matrices	O
are	O
the	O
Lie	O
algebra	O
of	O
the	O
rotation	O
group	O
this	O
elucidates	O
the	O
relation	O
between	O
three-space	O
,	O
the	O
cross	O
product	O
and	O
three-dimensional	O
rotations	O
.	O
</s>
<s>
Since	O
a	O
matrix	O
is	O
similar	B-Algorithm
to	O
its	O
own	O
transpose	O
,	O
they	O
must	O
have	O
the	O
same	O
eigenvalues	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
eigenvalues	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
always	O
come	O
in	O
pairs	O
±λ	O
(	O
except	O
in	O
the	O
odd-dimensional	O
case	O
where	O
there	O
is	O
an	O
additional	O
unpaired	O
0	O
eigenvalue	O
)	O
.	O
</s>
<s>
From	O
the	O
spectral	O
theorem	O
,	O
for	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
the	O
nonzero	O
eigenvalues	O
are	O
all	O
pure	O
imaginary	O
and	O
thus	O
are	O
of	O
the	O
form	O
where	O
each	O
of	O
the	O
are	O
real	O
.	O
</s>
<s>
Real	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
are	O
normal	B-Algorithm
matrices	I-Algorithm
(	O
they	O
commute	O
with	O
their	O
adjoints	B-Algorithm
)	O
and	O
are	O
thus	O
subject	O
to	O
the	O
spectral	O
theorem	O
,	O
which	O
states	O
that	O
any	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
can	O
be	O
diagonalized	O
by	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
eigenvalues	O
of	O
a	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
are	O
imaginary	O
,	O
it	O
is	O
not	O
possible	O
to	O
diagonalize	O
one	O
by	O
a	O
real	O
matrix	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
bring	O
every	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
to	O
a	O
block	B-Algorithm
diagonal	I-Algorithm
form	O
by	O
a	O
special	B-Algorithm
orthogonal	I-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
every	O
complex	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
can	O
be	O
written	O
in	O
the	O
form	O
where	O
is	O
unitary	O
and	O
has	O
the	O
block-diagonal	O
form	O
given	O
above	O
with	O
still	O
real	O
positive-definite	O
.	O
</s>
<s>
This	O
is	O
an	O
example	O
of	O
the	O
Youla	O
decomposition	O
of	O
a	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
defines	O
a	O
form	O
with	O
desirable	O
properties	O
for	O
vector	O
spaces	O
over	O
fields	O
of	O
characteristic	O
not	O
equal	O
to	O
2	O
,	O
but	O
in	O
a	O
vector	O
space	O
over	O
a	O
field	O
of	O
characteristic	O
2	O
,	O
the	O
definition	O
is	O
equivalent	O
to	O
that	O
of	O
a	O
symmetric	B-Algorithm
form	O
,	O
as	O
every	O
element	O
is	O
its	O
own	O
additive	O
inverse	O
.	O
</s>
<s>
A	O
bilinear	O
form	O
will	O
be	O
represented	O
by	O
a	O
matrix	O
such	O
that	O
,	O
once	O
a	O
basis	O
of	O
is	O
chosen	O
,	O
and	O
conversely	O
an	O
matrix	O
on	O
gives	O
rise	O
to	O
a	O
form	O
sending	O
to	O
For	O
each	O
of	O
symmetric	B-Algorithm
,	O
skew-symmetric	O
and	O
alternating	O
forms	O
,	O
the	O
representing	O
matrices	O
are	O
symmetric	B-Algorithm
,	O
skew-symmetric	O
and	O
alternating	O
respectively	O
.	O
</s>
<s>
Skew-symmetric	B-Algorithm
matrices	I-Algorithm
over	O
the	O
field	O
of	O
real	O
numbers	O
form	O
the	O
tangent	O
space	O
to	O
the	O
real	O
orthogonal	O
group	O
at	O
the	O
identity	O
matrix	O
;	O
formally	O
,	O
the	O
special	O
orthogonal	O
Lie	O
algebra	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
then	O
,	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
can	O
be	O
thought	O
of	O
as	O
infinitesimal	O
rotations	O
.	O
</s>
<s>
Another	O
way	O
of	O
saying	O
this	O
is	O
that	O
the	O
space	O
of	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
forms	O
the	O
Lie	O
algebra	O
of	O
the	O
Lie	O
group	O
The	O
Lie	O
bracket	O
on	O
this	O
space	O
is	O
given	O
by	O
the	O
commutator	O
:	O
</s>
<s>
It	O
is	O
easy	O
to	O
check	O
that	O
the	O
commutator	O
of	O
two	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
is	O
again	O
skew-symmetric	O
:	O
</s>
<s>
The	O
matrix	O
exponential	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
then	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
Lie	O
group	O
this	O
connected	O
component	O
is	O
the	O
special	O
orthogonal	O
group	O
consisting	O
of	O
all	O
orthogonal	B-Algorithm
matrices	I-Algorithm
with	O
determinant	O
1	O
.	O
</s>
<s>
Moreover	O
,	O
since	O
the	O
exponential	O
map	O
of	O
a	O
connected	O
compact	O
Lie	O
group	O
is	O
always	O
surjective	O
,	O
it	O
turns	O
out	O
that	O
every	O
orthogonal	B-Algorithm
matrix	I-Algorithm
with	O
unit	O
determinant	O
can	O
be	O
written	O
as	O
the	O
exponential	O
of	O
some	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
the	O
particular	O
important	O
case	O
of	O
dimension	O
the	O
exponential	O
representation	O
for	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
reduces	O
to	O
the	O
well-known	O
polar	O
form	O
of	O
a	O
complex	O
number	O
of	O
unit	O
modulus	O
.	O
</s>
<s>
The	O
exponential	O
representation	O
of	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
of	O
order	O
can	O
also	O
be	O
obtained	O
starting	O
from	O
the	O
fact	O
that	O
in	O
dimension	O
any	O
special	B-Algorithm
orthogonal	I-Algorithm
matrix	I-Algorithm
can	O
be	O
written	O
as	O
where	O
is	O
orthogonal	O
and	O
S	O
is	O
a	O
block	B-Algorithm
diagonal	I-Algorithm
matrix	O
with	O
blocks	O
of	O
order2	O
,	O
plus	O
one	O
of	O
order	O
1	O
if	O
is	O
odd	O
;	O
since	O
each	O
single	O
block	O
of	O
order	O
2	O
is	O
also	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
it	O
admits	O
an	O
exponential	O
form	O
.	O
</s>
<s>
Correspondingly	O
,	O
the	O
matrixS	O
writes	O
as	O
exponential	O
of	O
a	O
skew-symmetric	O
block	B-Algorithm
matrix	I-Algorithm
of	O
the	O
form	O
above	O
,	O
so	O
that	O
exponential	O
of	O
the	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
Conversely	O
,	O
the	O
surjectivity	O
of	O
the	O
exponential	O
map	O
,	O
together	O
with	O
the	O
above-mentioned	O
block-diagonalization	O
for	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
,	O
implies	O
the	O
block-diagonalization	O
for	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
More	O
intrinsically	O
(	O
i.e.	O
,	O
without	O
using	O
coordinates	O
)	O
,	O
skew-symmetric	O
linear	B-Architecture
transformations	I-Architecture
on	O
a	O
vector	O
space	O
with	O
an	O
inner	O
product	O
may	O
be	O
defined	O
as	O
the	O
bivectors	O
on	O
the	O
space	O
,	O
which	O
are	O
sums	O
of	O
simple	O
bivectors	O
(	O
2-blades	O
)	O
The	O
correspondence	O
is	O
given	O
by	O
the	O
map	O
where	O
is	O
the	O
covector	O
dual	O
to	O
the	O
vector	O
;	O
in	O
orthonormal	O
coordinates	O
these	O
are	O
exactly	O
the	O
elementary	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
An	O
matrix	O
is	O
said	O
to	O
be	O
skew-symmetrizable	O
if	O
there	O
exists	O
an	O
invertible	O
diagonal	B-Algorithm
matrix	I-Algorithm
such	O
that	O
is	O
skew-symmetric	O
.	O
</s>
