Linked from

The 24 pages that link to Hessian matrix, each with the reason it gives.

• GradientCompared with: The Hessian differentiates the gradient, capturing curvature rather than first-order change.

• Potential energy surfaceRelated: Its eigenvalues reveal local curvature and distinguish minima from saddle points.

• Jacobian matrixCompared with: It captures curvature, whereas a Jacobian records first-order change.

• Scalar fieldRelated: The Hessian gives a finer account of how the field bends near a point.

• Quadratic formRelated: The Hessian defines the local quadratic approximation used to classify critical points.

• Convex functionRelated: Positive semidefiniteness of the Hessian gives a multivariable convexity test.

• LaplacianRelated: The Laplacian is the trace of the Hessian in Euclidean coordinates.

• ConvexityRelated: A positive semidefinite Hessian establishes convexity on a convex domain.

• Fisher informationRelated: Likelihood curvature yields observed information and, in expectation, the information matrix.

• Morse theoryRelated: At a critical point, its eigenvalues determine nondegeneracy and the point’s Morse index.

• Symmetric matrixRelated: Under suitable smoothness conditions, mixed partial equality makes the Hessian symmetric.

• Positive-definite matrixRelated: A positive-definite Hessian at a stationary point establishes a strict local minimum.

• Second derivative testBroader topic: Its curvature information generalizes the one-variable second derivative test.

• Saddle pointRelated: Its eigenvalues identify upward and downward curvature at a stationary point.

• Maximum and minimumRelated: Its curvature information helps classify stationary points in several variables.

• Envelope theoremRelated: Second-order envelope formulas use curvature to capture changes in marginal value.

• Numerical optimizationRelated: Its curvature information can accelerate methods and characterize local optima.

• Second derivativeBroader topic: It generalizes the second derivative to describe curvature in several variables.

• Sylvester's law of inertiaRelated: At a critical point, its inertia determines the local quadratic behavior of the function.

• Sylvester's criterionRelated: A positive-definite Hessian certifies a strict local minimum, and Sylvester's criterion tests it.

• Nonlinear least squaresRelated: The exact objective curvature includes terms that Gauss–Newton omits.

• Function of several real variablesRelated: Its entries describe curvature and help classify stationary points.

• Jacobian matrix and determinantCompared with: The Hessian describes second-order variation, unlike the Jacobian’s first-order behavior.

• Symmetry of second derivativesRelated: Symmetry makes the Hessian symmetric when the mixed partials are continuous.