The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. The adjoint method formulates the gradient of a function towards its parameters in a constraint optimization form.

noun Mathematics. a square matrix obtained from a given square matrix and having the property that its product with the given matrix is equal to the determinant of the given matrix times the identity matrix.

the transpose of
Definition of adjoint : the transpose of a matrix in which each element is replaced by its cofactor.

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.

The adjoint method has long been considered as the tool of choice for gradient-based optimisation in computational fluid dynamics (CFD). It is the independence of the computational cost from the number of design variables that makes it particularly attractive for problems with large design spaces.

What is adjoint of a determinant?

The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. For a matrix A, the adjoint is denoted as adj (A). On the other hand, the inverse of a matrix A is that matrix which when multiplied by the matrix A give an identity matrix.

#### What is the adjoint of an operator?

In mathematics, the adjoint of an operator is a generalization of the notion of the Hermitian conjugate of a complex matrix to linear operators on complex Hilbert spaces. In this article the adjoint of a linear operator M will be indicated by M∗, as is common in mathematics.

#### What is adj a linear algebra?

In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix. The adjugate has sometimes been called the “adjoint”, but today the “adjoint” of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.