In the four-dimensional space of quaternions q = a + b i + c j + d k, the versors Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written U to indicate that the homogeneous factor multiplies on the left. The quaternion transform is The real and complex homographies described above are instances of the quaternion homography where θ is zero or π/2, respectively. Evidently the transform takes u → 0 → –1 and takes –u → ∞ → 1. Evaluating this homography at q = 1 maps the versoru into its axis: But Thus In this form the Cayley transform has been described as a rational parametrization of rotation: Let t = tan φ/2 in the complex number identity where the right hand side is the transform of t i and the left hand side represents the rotation of the plane by negative φ radians.
Inverse
Let Since where the equivalence is in the projective linear group over quaternions, the inverse of f is Since homographies are bijections, maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography f−1 produces rotations from the ball in ℝ3.
Matrix map
Among n×nsquare matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix. Then I + A is invertible, and the Cayley transform produces an orthogonal matrix, Q. 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. Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then is a skew-symmetric matrix. The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. A slightly different form is also seen, requiring different mappings in each direction: The mappings may also be written with the order of the factors reversed; however, A always commutes with −1, so the reordering does not affect the definition.
Examples
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. In the 3×3 case, we have where K = w2 + x2 + y2 + z2, and where w = 1. This we recognize as the rotation matrix corresponding to quaternion , except scaled so that w = 1 instead of the usual scaling so that w2 + x2 + y2 + z2 = 1. Thus vector 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.
Other matrices
We can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose is replaced by the conjugate transpose. This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, we may extend the definition further with choices of adjoint other than transpose or conjugate transpose. Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include −1. For example, we have We remark that A is skew-symmetric if and only ifQ is orthogonal with no eigenvalue −1.
Operator map
An infinite-dimensional version of an inner product space is a Hilbert space, and we can no longer speak of matrices. However, matrices are merely representations of linear operators, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators. Here the domain of U, dom U, is dom A. See self-adjoint operator for further details.
