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).

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:

Since ${\displaystyle \{\infty ,1,-1\}}$ is mapped to ${\displaystyle \{1,-i,i\}}$, and Möbius transformations permute the generalised circles in the complex plane, ${\displaystyle f}$ maps the real line to the unit circle. Furthermore, since ${\displaystyle f}$ is a homeomorphism and ${\displaystyle i}$ is taken to 0 by ${\displaystyle f}$, 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.

In the four-dimensional space of quaternions ${\displaystyle a+b{\vec {i}}+c{\vec {j}}+d{\vec {k}}}$, the versors