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Index of articles associated with the same name
Look up adjoint in Wiktionary, the free dictionary. In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism:
Adjoint
Dual to the Dirac spinor
In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved
Dirac_adjoint
Conjugate transpose of an operator in infinite dimensions
{\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the
Hermitian_adjoint
Relationship between two functors abstracting many common constructions
this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics
Adjoint_functors
Linear operator equal to its own adjoint
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot
Self-adjoint_operator
Mathematical term
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
Adjoint_representation
Linear differential equation
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect
Adjoint_equation
Result about when a matrix can be diagonalized
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral
Spectral_theorem
Element of *-algebra where x* equals x
mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a ∗ {\displaystyle a=a^{*}} ). Let A {\displaystyle
Self-adjoint_element
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though
Adjugate_matrix
In mathematics, an adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie
Adjoint_bundle
Numerical method
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It
Adjoint_state_method
In signal processing, the adjoint filter mask h ∗ {\displaystyle h^{*}} of a filter mask h {\displaystyle h} is reversed in time and the elements are
Adjoint_filter
Sub-field of logic programming
Multi-adjoint logic programming defines syntax and semantics of a logic programming program in such a way that the underlying maths justifying the results
Multi-adjoint logic programming
Multi-adjoint_logic_programming
Matrix operation which flips a matrix over its diagonal
resulting in an isomorphism between the transpose and adjoint of u. The matrix of the adjoint of a map is the transposed matrix only if the bases are
Transpose
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix A {\displaystyle
Conjugate_transpose
Matrix whose conjugate transpose is its negative (additive inverse)
thought of as skew-adjoint (since they are like 1 × 1 {\displaystyle 1\times 1} matrices), whereas real numbers correspond to self-adjoint operators. For
Skew-Hermitian_matrix
Induced map between the dual spaces of the two vector spaces
In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field,
Transpose_of_a_linear_map
Topological complex vector space
Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear
C*-algebra
Criteria in Category theory of Mathematics
mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the
Formal criteria for adjoint functors
Formal_criteria_for_adjoint_functors
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Linear operator defined on a dense linear subspace
the adjoint of T. It follows immediately from the above definition that the adjoint T ∗ {\displaystyle T^{*}} is closed. In particular, a self-adjoint operator
Unbounded_operator
Particular correspondence between two partially ordered sets
terminology encountered here is left adjoint (respectively right adjoint) for the lower (respectively upper) adjoint. An essential property of a Galois
Galois_connection
Type of vector space in math
This defines another bounded linear operator A* : H2 → H1, the adjoint of A. The adjoint satisfies A** = A. When the Riesz representation theorem is used
Hilbert_space
Typically linear operator defined in terms of differentiation of functions
self-adjoint operator is an operator equal to its own (formal) adjoint. If Ω is a domain in Rn, and P a differential operator on Ω, then the adjoint of
Differential_operator
Class of ordinary differential equations
differential equation (1) is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear homogenous ordinary differential equations
Sturm–Liouville_theory
Theorem about the dual of a Hilbert space
Self-adjoint operators A continuous linear operator A : H → H {\displaystyle A:H\to H} is called self-adjoint if it is equal to its own adjoint; that
Riesz_representation_theorem
Mathematical category
\operatorname {Presh} (D)} that admits a finite-limit-preserving left adjoint. C {\displaystyle C} is the category of sheaves on a Grothendieck site
Topos
Any entity that can be measured
c\in \mathbb {C} } . Observables are given by self-adjoint operators on V. Not every self-adjoint operator corresponds to a physically meaningful observable
Observable
Operation on self-adjoint operators
constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions
Extensions of symmetric operators
Extensions_of_symmetric_operators
General theory of mathematical structures
relationships. Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors
Category_theory
Graph representing edges of another graph
and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph. Hassler Whitney (1932) proved that with one
Line_graph
Measure used in functional analysis
function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM)
Projection-valued_measure
In mathematics, a linear operator acting on inner product space
to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically
Positive_operator
Square roots of the eigenvalues of the self-adjoint operator
non-negative) eigenvalues of the self-adjoint operator T ∗ T {\displaystyle T^{*}T} (where T ∗ {\displaystyle T^{*}} denotes the adjoint of T {\displaystyle T} )
Singular_value
Quasi-minuscule: 2n2–n (adjoint) E6 1, 27, 27. Quasi-minuscule: 78 (adjoint) E7 1, 56. Quasi-minuscule: 133 (adjoint) E8 1. Quasi-minuscule: 248 (adjoint) F4 1. Quasi-minuscule:
Minuscule_representation
Characterizing property of mathematical constructions
are then a pair of adjoint functors, with G {\displaystyle G} left-adjoint to F {\displaystyle F} and F {\displaystyle F} right-adjoint to G {\displaystyle
Universal_property
Functor type
representable if and only if it has a left adjoint. The categorical notions of universal morphisms and adjoint functors can both be expressed using representable
Representable_functor
Software for numerical solution of partial differential equations
discrete adjoint solvers. A distinguishing feature for researchers is its use of algorithmic differentiation (AD) to provide exact discrete adjoint sensitivities
SU2_code
Theorem in functional analysis
associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below. Let A be a n × n Hermitian matrix. As
Min-max_theorem
In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named
Friedrichs_extension
Partially ordered set in which all subsets have both a supremum and infimum
y\iff x\leq g(y)} where f is called the lower adjoint and g is called the upper adjoint. By the adjoint functor theorem, a monotone map between any pair
Complete_lattice
Categorical generalization of a function space in set theory
{\displaystyle (f\colon X\to Z)\mapsto (f^{Y}\colon X^{Y}\to Z^{Y})} , is a right adjoint to the product functor − × Y {\displaystyle -\times Y} . For this reason
Exponential_object
Mathematical concept
colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand
Limit_(category_theory)
contrat Gendarme Adjoint Maréchal-des-logis Gendarme Adjoint Brigadier Chef Gendarme Adjoint Brigadier Gendarme Adjoint première classe Gendarme Adjoint
Police_ranks_of_France
Category equipped with involution
for all morphisms f : A → B {\displaystyle f:A\to B} , there exists its adjoint f † : B → A {\displaystyle f^{\dagger }:B\to A} for all morphisms f {\displaystyle
Dagger_category
Mapping between categories
be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version
Functor
Property of some binary operations
the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator ad x : y ↦ [ x , y ] {\displaystyle \operatorname {ad} _{x}:y\mapsto
Jacobi_identity
Theorem relating unitary operators to one-parameter Lie groups
functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter
Stone's theorem on one-parameter unitary groups
Stone's_theorem_on_one-parameter_unitary_groups
Branch of functional analysis
the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann
Operator_algebra
Category theory constructs
category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M
Kan_extension
(on a complex Hilbert space) continuous linear operator
{\displaystyle N\colon H\rightarrow H} that commutes with its Hermitian adjoint N ∗ {\displaystyle N^{\ast }} , that is: N ∗ N = N N ∗ {\displaystyle N^{\ast
Normal_operator
Numerical calculations carrying along derivatives
in the adjoint; fanout in the primal causes addition in the adjoint; a unary function y = f(x) in the primal causes x̄ = ȳ f′(x) in the adjoint; etc. Reverse
Automatic_differentiation
Type of category in category theory
to C that maps objects X to X×Y and morphisms φ to φ × idY) has a right adjoint, usually denoted –Y, for all objects Y in C. For locally small categories
Cartesian_closed_category
Concept in category theory
see (Mac Lane 1997). As this is a fundamental example of adjoints, we spell it out: adjointness means that given a set X and an object (say, an R-module)
Forgetful_functor
133-dimensional exceptional simple Lie group
thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the
E7_(mathematics)
Theorem on boundedness of symmetric operators
operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem
Hellinger–Toeplitz_theorem
{\displaystyle V^{*}} is the adjoint of V. If T is a self-adjoint operator, then the compression T W {\displaystyle T_{W}} is also self-adjoint. When V is replaced
Compression (functional analysis)
Compression_(functional_analysis)
Mathematical conjecture about the Riemann zeta function
zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means
Hilbert–Pólya_conjecture
Paramilitary force in Vichy France
commander Chef régional adjoint Assistant regional commander Chef départemental Department commander Chef départemental adjoint Assistant department commander
Milice
Formulation of quantum mechanics on a Hilbert Space
observables of a quantum system are defined to be the (possibly unbounded) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } . A state
Dirac–von_Neumann_axioms
Federal and provincial police ranks in Canada
(directeur-adjoint) Lieutenant (lieutenant) Sergeant (sergent) Constable (agent) Director (directeur) Associate director (directeur-adjoint) Chief inspector
Police_ranks_of_Canada
Topological space
space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension.
Loop_space
operator for all B. E is called self-adjoint if E(B) is self-adjoint for all B. E is called spectral if it is self-adjoint and E ( B 1 ∩ B 2 ) = E ( B 1 )
Naimark's_dilation_theorem
Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product
Jordan_operator_algebra
Branch of functional analysis
Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. If T is a self-adjoint operator on a finite-dimensional
Borel_functional_calculus
Type of French boarding school
Principal(e) de collège Proviseur adjoint & Directeur/trice des études Proviseur adjoint Censeur Principal(e) adjoint(e) Économe Intendant(e) Inspectrice
Maison d'éducation de la Légion d'honneur
Maison_d'éducation_de_la_Légion_d'honneur
Concept in mathematical theory of categories
reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization. Dually, A is said
Reflective_subcategory
Optimal control equation
equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order
Costate_equation
correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real
State_(functional_analysis)
Special objects used in (mathematical) category theory
object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal
Initial_and_terminal_objects
President of France since 2017
2012). "Emmanuel Macron, un banquier d'affaires nommé secrétaire général adjoint de l'Elysée". Le Monde (in French). Archived from the original on 3 August
Emmanuel_Macron
In mathematics, invertible homomorphism
Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially
Isomorphism
Embedding of categories into functor categories
Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially
Yoneda_lemma
Generalization of monads
ordinary monads is that every adjoint pair of functors induces a monad, and that every monad is induced by an adjoint pair of functors. The two extremal
Pseudomonad_(category_theory)
Grand Unified Theory proposed in 1974
of the adjoint Higgs to be absorbed. The other real half acquires a mass coming from the D-terms. And the other three components of the adjoint Higgs,
Georgi–Glashow_model
symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n)
Quadratic_Lie_algebra
Association football club in Tunisia
https://www.mosaiquefm.net/fr/football/1486731/cab-bilel-ben-messaoud-entraineur-adjoint "Sport : Abdessalem Saidani, nouveau président du CA Bizertin" [Abdessalem
CA_Bizertin
Non-associative algebras with positive-definite quadratic form
R(a*) = R(a)*, so that the involution on the algebra corresponds to taking adjoints Re (ab) = Re (ba) if Re x = (x + x*)/2 = (x, 1)1 Re (ab)c = Re a(bc) L(a2)
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Militarised police force in France
Gendarme Adjoint Maréchal-des-logis Gendarme Adjoint Brigadier Chef Gendarme Adjoint Brigadier Gendarme Adjoint première classe Gendarme Adjoint Departmental
National_Gendarmerie
self-adjoint operator in ( A − ) 1 {\displaystyle (A^{-})_{1}} , then h {\displaystyle h} is in the strong-operator closure of the set of self-adjoint operators
Kaplansky_density_theorem
Governance position
French term for deputy mayor is maire-adjoint or adjoint au maire [fr]. The first deputy mayor is called premier adjoint. This term should not be confused
Deputy_mayor
Summability method in physics
particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics
Zeta_function_regularization
French politician (born 1987)
""Je vous écris de Vassivière" : la carte postale de Guillaume Guérin, adjoint aux finances du maire de Limoges". Le Dauphiné libéré (in French). 21 July
Guillaume_Guérin_(politician)
Mathematical use of "for all"
its domain. The left adjoint of this functor is the existential quantifier ∃ f {\displaystyle \exists _{f}} and the right adjoint is the universal quantifier
Universal_quantification
Relates 2 second-order elliptic operators on a manifold with the same principal symbol
symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle
Weitzenböck_identity
Theorem in algebra
module. Matlis duality can be conceptually explained using the language of adjoint functors and derived categories: the functor between the derived categories
Matlis_duality
Concept in quantum mechanics
The term "observable" has gained a technical meaning, denoting a self-adjoint operator that represents the possible results of a random variable. The
Observer_(quantum_physics)
Random matrix with gaussian entries
the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution
Gaussian_ensemble
Differential operator
In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues
Eta_invariant
Operator on a Hilbert space that shifts basis vectors
It can be regarded as an infinite-dimensional lower shift matrix. The adjoint of the unilateral shift, denoted S ∗ {\displaystyle S^{*}} , is the backward
Unilateral_shift_operator
CLAIREAUX Adjoints - Madame Josée QUEDINET-DETCHEVERRY Adjoints - Monsieur Frédéric BEAUMONT Adjoints - Monsieur Claude ARROSSAMENA Adjoints - Monsieur
Municipal governments in Saint Pierre and Miquelon
Municipal_governments_in_Saint_Pierre_and_Miquelon
Natural number
group SO(8). The special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent
8
Branch of mathematics
Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine
Nilpotent_Lie_algebra
Set of finitely supported functions from a group to a ring
Categorically, the group ring construction is left adjoint to "group of units"; the following functors are an adjoint pair: R [ − ] : G r p → R - A l g ( − ) ×
Group_ring
On boundary terms from integration by parts of a self-adjoint linear differential operator
gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville
Lagrange's identity (boundary value problem)
Lagrange's_identity_(boundary_value_problem)
Writing Lie algebra sets as matrices
ρ(X)(v). The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} on itself:
Lie_algebra_representation
Theorem in category theory
Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially
Lawvere's_fixed-point_theorem
Quotient of special unitary group by its center
of PU. The adjoint action of the infinite projective unitary group is useful in geometric definitions of twisted K-theory. Here the adjoint action of the
Projective_unitary_group
Eigenvalue transformation method
of the Lieb–Thirring inequality. The technique was developed for self-adjoint Schrödinger operators − Δ − V {\displaystyle -\Delta -V} on R n {\displaystyle
Birman–Schwinger_principle
ADJOINT
ADJOINT
ADJOINT
ADJOINT
Boy/Male
Muslim
Sovereign. Monarch.
Boy/Male
Greek
Broad.
Boy/Male
Indian, Persian, Tamil
Sweet
Boy/Male
Hindu, Indian, Kannada, Marathi, Telugu
Perfectly Hidden
Boy/Male
Indian
An Ornament Worn by Ancient Women
Boy/Male
American, Danish, French, German, Hebrew, Indian, Japanese, Swedish, Tamil
Lofty or Inspired; A Former Persian Province in Caucasus; Enlightened; Exalted; On High; A Prophet
Boy/Male
Danish, German, Polish
Bold; Brave
Boy/Male
British, English
Place Name; From the Oak Tree Meadow
Girl/Female
Indian
Heartedly
Girl/Female
Hindu, Indian
Reincarnated
ADJOINT
ADJOINT
ADJOINT
ADJOINT
ADJOINT
n.
An adjunct; a helper.
a.
Adjointing the shore.