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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 equation.
An adjoint state equation is introduced, including a new unknown variable. The adjoint method formulates the gradient of a function towards its parameters in a constraint optimization form. By using the dual form of this constraint optimization problem, it can be used to calculate the gradient very fast.
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 structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded ...
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [ 1 ] [ 2 ] It is occasionally known as adjunct matrix , [ 3 ] [ 4 ] or "adjoint", [ 5 ] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose .
In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule A x , y = x , A ∗ y , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}
One of Fredholm's theorems, closely related to the Fredholm alternative, concerns the existence of solutions to the inhomogeneous Fredholm equation (,) = ().Solutions to this equation exist if and only if the function () is orthogonal to the complete set of solutions {()} of the corresponding homogeneous adjoint equation:
Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as J μ = c ψ ¯ γ μ ψ {\displaystyle J^{\mu }=c{\bar {\psi }}\gamma ^{\mu }\psi } where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j :