Cayley transform
As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices.
In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikolski 1988).
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 On the upper half of the complex plane, the Cayley transform is:[1][2] Since
, and Möbius transformations permute the generalised circles in the complex plane,
maps the real line to the unit circle.
, the upper half-plane is mapped to the unit disk.
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.
, the versors Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written
In this form the Cayley transform has been described as a rational parametrization of rotation: Let
in the complex number identity[3] where the right hand side is the transform of
and the left hand side represents the rotation of the plane by negative
Since where the equivalence is in the projective linear group over quaternions, the inverse of
maps the vector quaternions to the 3-sphere of versors.
Then I + A is invertible, and the Cayley transform 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
In fact, Q must have determinant +1, so is special orthogonal.
[4] A slightly different form is also seen,[5][6] requiring different mappings in each direction, The mappings may also be written with the order of the factors reversed;[7][8] however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition.
In the 2×2 case, we have The 180° rotation matrix, −I, is excluded, though it is the limit as tan θ⁄2 goes to infinity.
This we recognize as the rotation matrix corresponding to quaternion (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).
Formally, the definition only requires some invertibility, so one can substitute for Q any matrix M whose eigenvalues do not include −1.
For example, Note that A is skew-symmetric (respectively, skew-Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1.
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.
