Search results
Results from the WOW.Com Content Network
The adjoint state space is chosen to simplify the physical interpretation of equation constraints. [3] Adjoint state techniques allow the use of integration by parts, resulting in a form which explicitly contains the physically interesting quantity. An adjoint state equation is introduced, including a new unknown variable. The adjoint method ...
[1] [2] It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations ˙ = where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.
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.
(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)
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.
Let ad X be the linear operator on g defined by ad X Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with Ad A for fixed A ∈ G the linear transformation of g given by Ad A Y = AYA −1. A standard combinatorial lemma which is utilized [18] in producing the above explicit expansions is given ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
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 ,}