<s>
In	O
mathematics	O
,	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
,	O
named	O
after	O
Arthur	O
Cayley	O
,	O
is	O
any	O
of	O
a	O
cluster	O
of	O
related	O
things	O
.	O
</s>
<s>
As	O
originally	O
described	O
by	O
,	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
a	O
mapping	O
between	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
and	O
special	B-Algorithm
orthogonal	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
transform	O
is	O
a	O
homography	B-Algorithm
used	O
in	O
real	O
analysis	O
,	O
complex	O
analysis	O
,	O
and	O
quaternionic	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
Hilbert	O
spaces	O
,	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
a	O
mapping	O
between	O
linear	B-Architecture
operators	I-Architecture
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
a	O
Cayley	B-Algorithm
transform	I-Algorithm
can	O
be	O
done	O
on	O
the	O
real	O
projective	O
line	O
.	O
</s>
<s>
The	O
Cayley	B-Algorithm
transform	I-Algorithm
here	O
will	O
permute	O
the	O
elements	O
of	O
{	O
1	O
,	O
0	O
,	O
−1	O
,	O
∞}	O
in	O
sequence	O
.	O
</s>
<s>
Thus	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
used	O
to	O
adapt	O
Legendre	O
polynomials	O
for	O
use	O
with	O
functions	O
on	O
the	O
positive	O
real	O
numbers	O
with	O
Legendre	O
rational	O
functions	O
.	O
</s>
<s>
On	O
the	O
Riemann	O
sphere	O
,	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
:	O
</s>
<s>
In	O
terms	O
of	O
the	O
models	O
of	O
hyperbolic	O
geometry	O
,	O
this	O
Cayley	B-Algorithm
transform	I-Algorithm
relates	O
the	O
Poincaré	O
half-plane	O
model	O
to	O
the	O
Poincaré	O
disk	O
model	O
.	O
</s>
<s>
In	O
electrical	O
engineering	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
has	O
been	O
used	O
to	O
map	O
a	O
reactance	O
half-plane	O
to	O
the	O
Smith	B-Application
chart	I-Application
used	O
for	O
impedance	O
matching	O
of	O
transmission	O
lines	O
.	O
</s>
<s>
The	O
real	O
and	O
complex	O
homographies	B-Algorithm
described	O
above	O
are	O
instances	O
of	O
the	O
quaternion	O
homography	B-Algorithm
where	O
is	O
zero	O
or	O
,	O
respectively	O
.	O
</s>
<s>
Evaluating	O
this	O
homography	B-Algorithm
at	O
maps	O
the	O
versor	O
into	O
its	O
axis	O
:	O
</s>
<s>
Since	O
homographies	B-Algorithm
are	O
bijections	B-Algorithm
,	O
maps	O
the	O
vector	O
quaternions	O
to	O
the	O
3-sphere	O
of	O
versors	O
.	O
</s>
<s>
As	O
versors	O
represent	O
rotations	O
in	O
3-space	O
,	O
the	O
homography	B-Algorithm
produces	O
rotations	O
from	O
the	O
ball	O
in	O
.	O
</s>
<s>
Among	O
n×n	O
square	B-Algorithm
matrices	I-Algorithm
over	O
the	O
reals	O
,	O
with	O
I	O
the	O
identity	O
matrix	O
,	O
let	O
A	O
be	O
any	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
(	O
so	O
that	O
AT	O
=	O
−A	O
)	O
.	O
</s>
<s>
produces	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
Q	O
(	O
so	O
that	O
QTQ	O
=	O
I	O
)	O
.	O
</s>
<s>
is	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
(	O
See	O
also	O
:	O
Involution	B-Algorithm
.	O
)	O
</s>
<s>
The	O
condition	O
on	O
Q	O
automatically	O
excludes	O
matrices	B-Architecture
with	O
determinant	O
−1	O
,	O
but	O
also	O
excludes	O
certain	O
special	B-Algorithm
orthogonal	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
for	O
some	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
A	O
and	O
some	O
diagonal	O
matrix	O
E	O
with	O
±1	O
as	O
entries	O
.	O
</s>
<s>
Again	O
excluded	O
are	O
180°	O
rotations	O
,	O
which	O
in	O
this	O
case	O
are	O
all	O
Q	O
which	O
are	O
symmetric	B-Algorithm
(	O
so	O
that	O
QT	O
=	O
Q	O
)	O
.	O
</s>
<s>
One	O
can	O
extend	O
the	O
mapping	O
to	O
complex	O
matrices	B-Architecture
by	O
substituting	O
"	O
unitary	B-Algorithm
"	O
for	O
"	O
orthogonal	O
"	O
and	O
"	O
skew-Hermitian	O
"	O
for	O
"	O
skew-symmetric	O
"	O
,	O
the	O
difference	O
being	O
that	O
the	O
transpose	O
( ·T	O
)	O
is	O
replaced	O
by	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
( ·H	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
one	O
may	O
extend	O
the	O
definition	O
further	O
with	O
choices	O
of	O
adjoint	O
other	O
than	O
transpose	O
or	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
A	O
is	O
skew-symmetric	O
(	O
respectively	O
,	O
skew-Hermitian	O
)	O
if	O
and	O
only	O
if	O
Q	O
is	O
orthogonal	O
(	O
respectively	O
,	O
unitary	B-Algorithm
)	O
with	O
no	O
eigenvalue	O
−1	O
.	O
</s>
<s>
An	O
infinite-dimensional	O
version	O
of	O
an	O
inner	O
product	O
space	O
is	O
a	O
Hilbert	O
space	O
,	O
and	O
one	O
can	O
no	O
longer	O
speak	O
of	O
matrices	B-Architecture
.	O
</s>
<s>
However	O
,	O
matrices	B-Architecture
are	O
merely	O
representations	O
of	O
linear	B-Architecture
operators	I-Architecture
,	O
and	O
these	O
can	O
be	O
used	O
.	O
</s>
<s>
So	O
,	O
generalizing	O
both	O
the	O
matrix	O
mapping	O
and	O
the	O
complex	O
plane	O
mapping	O
,	O
one	O
may	O
define	O
a	O
Cayley	B-Algorithm
transform	I-Algorithm
of	O
operators	O
.	O
</s>
