adjoint

English

Etymology

From French adjoindre (“to join”), from late 19th C; see also adjoin. Doublet of 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.

Pronunciation

• IPA^(key): /ˈæd͡ʒ.ɔɪnt/

• Audio (Southern England):

  (file)

Adjective

adjoint (not comparable)

1. (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).
2. (mathematics, category theory, of a functor) That is related to another functor by an adjunction.
3. (geometry, of one curve to another curve) Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
   • 1933, H. F. Baker, Principles of Geometry, volume 5, published 2010, page 103:

     The sets A + A₀, B + B₀, together, form the complete intersection, with f = 0, of a composite adjoint curve of order m + k, consisting of the adjoint curve of order m through A + B, together with the non-adjoint curve ω = 0; and the set B + B₀ consists of p points, and lies on i + j adjoint φ-curves of f = 0.

   • 1963, Julian Lowell Coolidge, A History of Geometrical Methods, page 205:

     As we have stated before, a curve $f'$ is adjoint to a curve $f$ if it have at least the multiplicity $r_{i}-1$ at each point where $f$ has the multiplicity $r_{i}$. A first polar $\sum _{i}y_{i}\left(\partial f/\partial x_{i}\right)=0$ is an example of an adjoint curve.

   • 2016, Eugene Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, page 216:

     This imposes n(n - 3)/2 conditions on the n-gon adjoint curve.

Usage notes

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

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)

Derived terms

• adjointable
• adjoint action
• adjoint curve
• adjoint endomorphism
• adjoint functor
• adjointly
• adjoint matrix
• adjointness
• adjoint operator
• adjoint representation
• left adjoint functor
• nonadjoint
• right adjoint functor

 

Related terms

• adjunction (noun)
• coadjoint
• self-adjoint
• sub-adjoint

Translations

matrix

• Esperanto: adjunkta
• German: adjungiert (de)
• Spanish: adjunta (es) f

Noun

adjoint (plural adjoints)

1. (mathematics) The transpose of the cofactor matrix of a given square matrix.
2. (mathematics, linear algebra, of a matrix) Transpose conjugate.
3. (mathematics, mathematical analysis, of an operator) Hermitian conjugate.
4. (mathematics, category theory) A functor related to another functor by an adjunction.
5. (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.
6. An assistant to someone who holds a position in the military or civil service.
   • 1798, Theobald Wolfe Tone, The Writings of Theobald Wolfe Tone 1763-98, Volume 3:

     . Nominated the citizens Fayolles, captain of infantry, and Favery of the Engineers to be my adjoints, and dispatched the letters of nomination to the Minister at War, so now I am fairly afloat.

   • 1884, William Morris, Art and Socialism:

     Even as it is, by the reckless aggregation of machine-workers and their adjoints in the great cities and the manufacturing districts, it has kept down life amongst us, and keeps it down to a miserably low standard; so low that any standpoint for improvement is hard to think of even.

   • 1975, Elizabeth L. Saxe, Sidney Wilfred Mintz, Working Papers in Haitian Society and Culture, page 60:

     The chef and his adjoints are seen by this code as part of the military, subject to military jurisdiction despite their obligation to execute the orders of the administrative council and of the justice of the peace.

7. An assistant mayor of a French commune.
   • 1929, Chester Collins Maxey, Urban Democracy, page 89:

     A vigorous, aggressive, and ambitious mayor will not rely overmuch upon his adjoints, but a mayor of the more passive type will lean heavily upon them.

   • 1989, Alan Forrest, Conscripts and Deserters, →ISBN:

     Mayors and their adjoints were often simple cultivateurs like the majority of the community.

Derived terms

• biadjoint
• sub-adjoint

Translations

matrix

• German: Adjungierte f
• Spanish: matriz de adjuntos f, matriz de cofactores f

matrix

• Bulgarian: ^{(please verify)} спрегнат (bg) (spregnat), ^{(please verify)} свързан (bg) (svǎrzan)
• Slovak: ^{(please verify)} adjungovaný

References

• Adjoint on Wolfram MathWorld

French

Pronunciation

• IPA^(key): /ad.ʒwɛ̃/

• Audio:

  (file)

Noun

adjoint m (plural adjoints)

1. deputy, assistant
2. (linguistics) adjunct

Participle

adjoint (feminine adjointe, masculine plural adjoints, feminine plural adjointes)

1. past participle of adjoindre

Further reading

• “adjoint”, in Trésor de la langue française informatisé [Digitized Treasury of the French Language], 2012.