Cayley

Cayley is a surname and place name, most notably associated with Arthur Cayley, a 19th-century English mathematician. He contributed to the development of group theory, algebra, and geometry, and his work laid the groundwork for modern algebra. The surname has been attached to several mathematical constructions and objects named in his honor.

Cayley graph: a graph that encodes the structure of a group given a particular generating set. Each vertex corresponds to a group element, and directed edges represent multiplication by generators. The Cayley graph is a central tool in geometric group theory and the study of symmetries.

Cayley–Hamilton theorem: every square matrix over a commutative ring satisfies its own characteristic equation; equivalently, p(A) = 0 where p is the characteristic polynomial of A. This theorem provides a fundamental link between linear transformations and polynomials.

Cayley transform: a Möbius transformation used in complex analysis and operator theory to map between domains such as the unit disk and the right half-plane, aiding the study of spectral properties and conformal mappings.

Cayley–Dickson construction: a recursive process to create algebras from existing ones, yielding complex numbers from real numbers, quaternions from complex numbers, octonions from quaternions, and beyond. The construction is named for Arthur Cayley and Leonard Dickson and is central to the study of higher-dimensional algebras.

Cayley table: the multiplication table of a finite group, used to summarize the group operation and analyze structure; named after Cayley.

Cayley numbers: another name for octonions, an eight-dimensional, non-associative normed division algebra over the real numbers; associated with Cayley.

Cayley, Alberta: a small village in southern Alberta, Canada.