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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 ...
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.
There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending ...
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
bordered Hessian An alternative name for the reciprocant bracket An invariant given by either the pairing of a vector and a vector in the dual space, or the determinant of a matrix form by n vectors of an n-dimensional space (in other words their exterior product in the top exterior power). Brioschi covariant
The Hessian, however is not invertible, as its rank is 3N-6 (6 variables responsible to a rigid body motion). In other words, the eigen values corresponding to the rigid motion are 0, resulting in the determinant being 0, making the matrix not invertible. To obtain a pseudo inverse, a solution to the eigenvalue problem is obtained:
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 ...
A form f is itself a covariant of degree 1 and order n.. The discriminant of a form is an invariant.. The resultant of two forms is a simultaneous invariant of them.. The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix