In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by, the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators .
A simple example of a Cayley transform can be done on the real projective line. The Cayley transform here will permute the elements of in sequence. For example, it maps the positive real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions.
As a real homography, points are described with projective coordinates, and the mapping is
[y, 1]=\left[
| x-1 | |
| x+1 |
, 1\right]\thicksim[x-1, x+1]=[x, 1]\begin{pmatrix}1&1\ -1&1\end{pmatrix}.
On the upper half of the complex plane, the Cayley transform is:[1]
f(z)=
| z-i | |
| z+i |
.
\{infty,1,-1\}
\{1,-i,i\}
f
f
i
f
In terms of the models of hyperbolic geometry, this Cayley transform relates the Poincaré half-plane model to the Poincaré disk model.
In electrical engineering the Cayley transform has been used to map a reactance half-plane to the Smith chart used for impedance matching of transmission lines.
In the four-dimensional space of quaternions
a+b\vec{i}+c\vec{j}+d\vec{k}
u(\theta,r)=\cos\theta+r\sin\theta
Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written
U[a,b]
f(u,q)=U[q,1]\begin{pmatrix}1&1\ -u&u\end{pmatrix}=U[q-u, q+u]\simU[(q+u)-1(q-u), 1].
The real and complex homographies described above are instances of the quaternion homography where
\theta
\pi/2
u\to0\to-1
-u\toinfty\to1
Evaluating this homography at
q=1
u
f(u,1)=(1+u)-1(1-u)=(1+u)*(1-u)/|1+u|2.
|1+u|2=(1+u)(1+u*)=2+2\cos\theta, and (1+u*)(1-u)=-2r\sin\theta.
Thus
f(u,1)=-r
| \sin\theta | |
| 1+\cos\theta |
=-r\tan
| \theta | |
| 2 |
.
In this form the Cayley transform has been described as a rational parametrization of rotation: Let
t=\tan\phi/2
e-i=
| 1-ti | |
| 1+ti |
ti
\phi
Let
u*=\cos\theta-r\sin\theta=u-1.
\begin{pmatrix}1&1\ -u&u\end{pmatrix} \begin{pmatrix}1&-u*\ 1&u*\end{pmatrix} = \begin{pmatrix}2&0\ 0&2\end{pmatrix} \sim \begin{pmatrix}1&0\ 0&1\end{pmatrix} ,
f(u,1)
U[p,1]\begin{pmatrix}1&-u*\ 1&u*\end{pmatrix} = U[p+1, (1-p)u*]\simU[u(1-p)-1(p+1), 1].
f-1(u,1)
f-1
\R3
Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that AT = −A).
Then I + A is invertible, and the Cayley transform
Q=(I-A)(I+A)-1
Q=(I+A)-1(I-A)
Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then
A=(I-Q)(I+Q)-1
However, any rotation (special orthogonal) matrix Q can be written as
Q=l((I-A)(I+A)-1r)2
for some skew-symmetric matrix A; more generally any orthogonal matrix Q can be written as
Q=E(I-A)(I+A)-1
for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries.[3]
A slightly different form is also seen,[4] requiring different mappings in each direction,
\begin{align} Q&=(I-A)-1(I+A),\\[5mu] A&=(Q-I)(Q+I)-1. \end{align}
The mappings may also be written with the order of the factors reversed;[5] [6] however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition.
In the 2×2 case, we have
\begin{bmatrix}0&\tan
| \theta | |
| 2 |
\ -\tan
| \theta | |
| 2 |
&0\end{bmatrix} \leftrightarrow \begin{bmatrix}\cos\theta&-\sin\theta\ \sin\theta&\cos\theta\end{bmatrix}.
In the 3×3 case, we have
\begin{bmatrix}0&z&-y\ -z&0&x\ y&-x&0\end{bmatrix} \leftrightarrow
| 1 | |
| K |
\begin{bmatrix} w2+x2-y2-z2&2(xy-wz)&2(wy+xz)\\ 2(xy+wz)&w2-x2+y2-z2&2(yz-wx)\\ 2(xz-wy)&2(wx+yz)&w2-x2-y2+z2 \end{bmatrix},
where K = w2 + x2 + y2 + z2, and where w = 1. This we recognize as the rotation matrix corresponding to quaternion
w+ix+jy+kz
(by a formula Cayley had published the year before), except scaled so that w = 1 instead of the usual scaling so that w2 + x2 + y2 + z2 = 1. Thus vector (x,y,z) is the unit axis of rotation scaled by tan θ⁄2. Again excluded are 180° rotations, which in this case are all Q which are symmetric (so that QT = Q).
One can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·T) is replaced by the conjugate transpose (·H). This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, one may extend the definition further with choices of adjoint other than transpose or conjugate transpose.
Formally, the definition only requires some invertibility, so one can substitute for Q any matrix M whose eigenvalues do not include −1. For example,
\begin{bmatrix}0&-a&ab-c\ 0&0&-b\ 0&0&0\end{bmatrix} \leftrightarrow \begin{bmatrix}1&2a&2c\ 0&1&2b\ 0&0&1\end{bmatrix}.
An infinite-dimensional version of an inner product space is a Hilbert space, and one can no longer speak of matrices. However, matrices are merely representations of linear operators, and these can be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators.
\begin{align} U&{}=(A-iI)(A+iI)-1\\ A&{}=i(I+U)(I-U)-1\end{align}