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The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...
Recover the Hessian matrix from the compact matrix. Steps one and two need only be carried out once, and tend to be costly. When one wants to calculate the Hessian at numerous points (such as in an optimization routine), steps 3 and 4 are repeated.
The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. [1] [2]
In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial ...
There also exist various quasi-Newton methods, where an approximation for the Hessian (or its inverse directly) is built up from changes in the gradient. If the Hessian is close to a non-invertible matrix, the inverted Hessian can be numerically unstable and the solution may diverge. In this case, certain workarounds have been tried in the past ...
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In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.