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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. In complex analysis, the Cayley transform is a conformal mapping (Rudin 1987) in which the image of the upper complex half-plane is the unit disk (Remmert 1991, pp. 82ff, 275).