Hessian

The Hessian, in mathematics, refers to the Hessian matrix, a concept named after the German mathematician Otto Hesse. For a scalar-valued function f of several variables, f(x1, x2, ..., xn), the Hessian H(f) is the n×n matrix of second-order partial derivatives, with entries Hij = ∂²f/∂xi∂xj. If f is twice continuously differentiable, the Hessian is symmetric due to Clairaut’s theorem, and it provides second-order information about the function’s curvature.

The Hessian appears in the second-order Taylor expansion of f around a point and is central to local optimization. At a critical point where the gradient vanishes, the eigenvalues of the Hessian determine the point’s nature: a positive definite Hessian indicates a local minimum, a negative definite Hessian a local maximum, and an indefinite Hessian a saddle point. If the Hessian is positive or negative semidefinite or singular, the test may be inconclusive. The determinant of the Hessian equals the product of its eigenvalues and gives a global curvature measure, while principal minors (Sylvester’s criterion) provide a more nuanced definiteness test.

The term also extends into differential geometry as the covariant Hessian on manifolds, which generalizes the concept using the Levi-Civita connection. Beyond mathematics, Hessian relates to Hesse, the German state of Hesse (Hessen in German), from which demonyms such as Hessian are derived, and to Hessians, historical mercenaries from Hesse-Cassel who served in various wars.

In summary, the Hessian captures second-order behavior of functions, aiding curvature analysis, Taylor approximations, and optimization.