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According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : R n → R n is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at a point p is
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ((+)) < A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute ...
For a system of the form (,, ′) =, some sources also require that the Jacobian matrix (,,) be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system.
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.
The Jacobian of this system is the bordered matrix [ F u F λ u ˙ ∗ λ ˙ ] {\displaystyle \left[{\begin{array}{cc}F_{u}&F_{\lambda }\\{\dot {u}}^{*}&{\dot {\lambda }}\\\end{array}}\right]} At regular points, where the unmodified Jacobian is full rank, the tangent vector spans the null space of the top row of this new Jacobian.
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; Jacobi coordinates, a simplification of coordinates for an n-body system; Jacobi identity for non-associative binary ...
That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances ...
A Hermitian matrix, H is defined by the conjugate transpose symmetry property: † = , =, By definition, the complex conjugate of a complex unitary rotation matrix, R is its inverse and also a complex unitary rotation matrix: