Adjoint Explained

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 in case of matrices
• Hermitian adjoint (adjoint of a linear operator) in functional analysis
• Adjoint endomorphism of a Lie algebra
• Adjoint representation of a Lie group
• Adjoint functors in category theory
• Adjunction (field theory)
• Adjunction formula (algebraic geometry)
• Adjunction space in topology
• Conjugate transpose of a matrix in linear algebra
• Adjugate matrix, related to its inverse
• Adjoint equation
• The upper and lower adjoints of a Galois connection in order theory
• The adjoint of a differential operator with general polynomial coefficients
• Kleisli adjunction
• Monoidal adjunction
• Quillen adjunction
• Axiom of adjunction in set theory
• Adjunction (rule of inference)