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The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. [1] It has applications in geophysics, seismic imaging, photonics and more recently in neural networks. [2] The adjoint state space is chosen to simplify the physical interpretation of equation ...
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 addition to support for vectorized arithmetic and relational operations, these languages also vectorize common mathematical functions such as sine. For example, if x is an array, then y = sin (x) will result in an array y whose elements are sine of the corresponding elements of the array x. Vectorized index operations are also supported.
In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.
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
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
The operator norm is also compatible with the composition, or multiplication, of operators: if , and are three normed spaces over the same base field, and : and : are two bounded operators, then it is a sub-multiplicative norm, that is: ‖ ‖ ‖ ‖ ‖ ‖.
In modern JavaScript it's considered bad form to use the Array type as an associative array. Consensus is that the Object type and Map / WeakMap classes are best for this purpose. The reasoning behind this is that if Array is extended via prototype and Object is kept pristine, for and for-in loops will work as expected on associative 'arrays'.