Definitions For Adjoint

Noun

ADJOINT (plural ADJOINTs) (mathematics) The transpose of the cofactor matrix of a given square matrix. (mathematics, linear algebra, of a matrix) transpose conjugate, 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.

Derived terms

{{der-top, }} biadjoint {{der-mid}} sub-adjoint {{der-bottom}}

Translations

German: Adjungierte, f Spanish: matriz de adjuntos, f, matriz de cofactores, f {{rfc, en, Are these adjectives or nouns?
(German adjungiert is an adjective and Spanish matriz de .. f is noun. Both were once listed here. The Slovak term in -ý maybe could denote an adjective too. The Bulgarian свързан is an adjective (прилагателно име) according to the Bulgarian wiktionary.)}} Bulgarian: {{t+check, bg, спрегнат}}, {{t+check, bg, свързан}} Slovak: {{t-check, sk, adjungovaný}}

Adjective

(not comparable) (mathematics) {{non-gloss definition, Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugate, 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 {{pedialite, Hermitian adjoint}}) In the case of an {{w, Adjoint representation, 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 differentiation, differentiating ("linearising") the group action of conjugation (i.e., differentiating the function x → gxg-1 for each element g). The {{w, Adjoint representation of a Lie algebra, adjoint}} representation of a lie algebra, 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 {{qualifier, in certain contexts}}

Derived terms

{{der-top}} adjoint action adjoint curve adjoint endomorphism adjoint functor {{der-mid}} adjoint matrix adjoint operator adjoint representation left adjoint functor right adjoint functor {{der-bottom}}

Related terms

adjunction {{qualifier, noun}} coadjoint self-adjoint sub-adjoint

Translations

German: adjungiert

Etymology

From {{der, en, fr, adjoindre, , to join}}, from late 19th C; see also adjoin. {{doublet, en, adjunct}}. In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.