ADJOINT

Etymology

Adjective

adjoint (not comparable)

(mathematics) Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugation (either inner automorphism or complex conjugation).

(mathematics, category theory, of a functor) That is related to another functor by an adjunction.

(geometry, of one curve to another curve) Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.

Usage notes

The adjoint operator, or Hermitian transpose, of an operator generalises the concept of transpose conjugate of a matrix. (See )

In the case of an adjoint representation of a Lie group, the representation in question describes the group's elements as linear transformations of its Lie algebra, itself considered as a vector space. The representation is obtained by differentiating ("linearising") the group action of conjugation (i.e, differentiating the function x → gxg^-1 for each element g).

The adjoint representation of a Lie algebra is the differential of the adjoint representation of a Lie group at the identity element of the group.

In relation to functors in category theory (and therefore in numerous fields of mathematics), the term is often synonymous with adjunct and the functors are said to be related by an adjunction. Functors may be left or right adjoint (adjunct).

Synonyms

• (mathematics): adjunct (in certain contexts)

Noun

adjoint (plural adjoints)

(mathematics) The transpose of the cofactor matrix of a given square matrix.

(mathematics, linear algebra, of a matrix) Transpose conjugate.

(mathematics, analysis, of an operator) Hermitian conjugate.

(mathematics, category theory) A functor related to another functor by an adjunction.

(geometry, algebraic geometry) A curve A such that any point of a given curve C of multiplicity r has multiplicity at least r–1 on A. Sometimes the multiple points of C are required to be ordinary, and if this condition is not satisfied the term sub-adjoint is used.

An assistant to someone who holds a position in the military or civil service.

An assistant mayor of a French commune.

Source: Wiktionary

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Ad"joint, n.

Definition: An adjunct; a helper. [Obs.]

Source: Webster’s Unabridged Dictionary 1913 Edition

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