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In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoint or adjunction may mean: * Adjoint of a linear map, also called its transpose *
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
(adjoint of a linear operator) in functional analysis *
Adjoint endomorphism In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of a Lie algebra *
Adjoint representation of a Lie group In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
*
Adjoint functors In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
in category theory *
Adjunction (field theory) In mathematics, particularly in algebra, a field extension is a pair of Field (mathematics), fields E\subseteq F, such that the operations of ''E'' are those of ''F'' Restriction (mathematics), restricted to ''E''. In this case, ''F'' is an extens ...
* Adjunction formula (algebraic geometry) *
Adjunction space In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ' ...
in topology *
Conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of a matrix in linear algebra *
Adjugate matrix In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...
, related to its inverse *
Adjoint equation An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equat ...
* The upper and lower adjoints of a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
in order theory * The adjoint of a differential operator with general polynomial coefficients *
Kleisli adjunction In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise f ...
*
Monoidal adjunction Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet,J) are two monoidal categories. A monoidal adjunction between two lax monoidal functors :(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) and (G,n):(\mathcal D,\bullet,J)\to(\mathc ...
*
Quillen adjunction In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the ...
*
Axiom of adjunction In mathematical set theory, the axiom of adjunction states that for any two sets ''x'', ''y'' there is a set ''w'' = ''x'' ∪  given by "adjoining" the set ''y'' to the set ''x''. : \forall x \,\forall y \,\exists w \,\forall z ...
in set theory * Adjunction (rule of inference) {{sia, mathematics Mathematical terminology