In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), 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.
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In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), 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 (Nikolski 1988).
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Real homography
A simple example of a Cayley transform can be done on the real projective line. The Cayley transform here will permute the elements of {1, 0, −1, ∞} 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
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
{\displaystyle Q=(I-A)(I+A)^{-1}\,\!}
produces an orthogonal matrix, Q (so that QTQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as
The 180° rotation matrix, −I, is excluded, though it is the limit as tan θ⁄2 goes to infinity.
Other matrices
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,
[
0
−
a
a
b
−
c
0
0
−
b
0
0
0
]
↔
[
1
2
a
2
c
0
1
2
b
0
Operator map
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.
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, and Möbius transformations permute the generalised circles in the complex plane,
f
{\displaystyle f}
maps the real line to the unit circle. Furthermore, since
f
{\displaystyle f}
is a homeomorphism and
i
{\displaystyle i}
is taken to 0 by
f
{\displaystyle f}
, the upper half-plane is mapped to the unit disk.
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.
U
[
a
,
b
]
{\displaystyle U[a,b]}
to indicate that the homogeneous factor multiplies on the left. The quaternion transform is
maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography
f
−
1
{\displaystyle f^{-1}}
produces rotations from the ball in
R
3
{\displaystyle \mathbb {R} ^{3}}
.
)
{\displaystyle Q=(I+A)^{-1}(I-A)}
. In fact, Q must have determinant +1, so is special orthogonal.
Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then
A
=
(
I
−
Q
)
(
I
+
Q
)
−
1
{\displaystyle A=(I-Q)(I+Q)^{-1}\,\!}
is a skew-symmetric matrix. (See also: Involution.) The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.
However, any rotation (special orthogonal) matrix Q can be written as
The mappings may also be written with the order of the factors reversed; however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition.
(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).
