Adjoint
Encyclopedia
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 = (x, By).
Specifically, adjoint may mean:
Specifically, adjoint may mean:
- Hermitian adjointHermitian adjointIn mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...
(adjoint of a linear operator) in functional analysis - Adjoint functorsAdjoint functorsIn mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
in category theory - Adjoint representation of a Lie group
- Adjoint endomorphismAdjoint endomorphismIn mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups....
of a Lie algebra - Conjugate transposeConjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of a matrix in linear algebra - Adjugate matrixAdjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix is a matrix that plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions....
, related to its inverse - Adjoint equationAdjoint equationAn 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 equation. Methods based on solution of adjoint...
- The upper and lower adjoints of a Galois connectionGalois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
in order theory - For the adjoint of a differential operator with general polynomial coefficients see differential operator