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  2. Costate equation - Wikipedia

    en.wikipedia.org/wiki/Costate_equation

    [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.

  3. Adjoint state method - Wikipedia

    en.wikipedia.org/wiki/Adjoint_state_method

    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 ...

  4. Adjugate matrix - Wikipedia

    en.wikipedia.org/wiki/Adjugate_matrix

    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.

  5. Adjoint - Wikipedia

    en.wikipedia.org/wiki/Adjoint

    (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)

  6. Adjoint equation - Wikipedia

    en.wikipedia.org/wiki/Adjoint_equation

    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.

  7. Associated Legendre polynomials - Wikipedia

    en.wikipedia.org/wiki/Associated_Legendre...

    This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial .

  8. Jacobi's formula - Wikipedia

    en.wikipedia.org/wiki/Jacobi's_formula

    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

  9. Hermitian adjoint - Wikipedia

    en.wikipedia.org/wiki/Hermitian_adjoint

    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 ,}