Cayley transform

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 (Nikol’skii 2001). 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 (Nikol’skii 2001). The Cayley transform is an automorphism of the real projective line that permutes the elements of {1, 0, −1, ∞} in sequence. For example, it maps the positive real numbers to the interval . 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 In the complex projective plane the Cayley transform is: Since {∞, 1, –1 } is mapped to {1, –i, i }, and Möbius transformations permute the generalised circles in the complex plane, f maps the real line to the unit circle. Furthermore, since f is continuous and i is taken to 0 by 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. 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(a,b) 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.