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If m = n, then f is a function from R n to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point.
Jacobi matrix may refer to: Jacobian matrix and determinant of a smooth map between Euclidean spaces or smooth manifolds Jacobi operator (Jacobi matrix), a tridiagonal symmetric matrix appearing in the theory of orthogonal polynomials
The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. [ 9 ] [ 10 ] A further generalization for a function between Banach spaces is the Fréchet derivative .
Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden ...
The complex torus associated to a genus algebraic curve, obtained by quotienting by the lattice of periods is referred to as the Jacobian variety. This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic ...
Jacobian matrix and determinant; This page is a redirect. The following categories are used to track and monitor this redirect: From a more specific name: ...
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.
Jacobian matrix and determinant of a smooth map between Euclidean spaces or smooth manifolds; Jacobi operator (Jacobi matrix), a tridiagonal symmetric matrix appearing in the theory of orthogonal polynomials; Jacobi polynomials, a class of orthogonal polynomials; Jacobi symbol, a generalization of the Legendre symbol