Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

Specifically, suppose f : ℝⁿ → ℝ is a function taking as input a vector x ∈ ℝⁿ and outputting a scalar f(x) ∈ ℝ; if all second partial derivatives of f exist and are continuous over the domain of the function, then the Hessian matrix H of f is a square n×n matrix, usually defined and arranged as follows:

or, component-wise:

The determinant of the above matrix is also sometimes referred to as the Hessian.

The Hessian matrix can be considered related to the Jacobian matrix by H(f(x)) = J(∇f(x))^T.

The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Schwarz's theorem). For example,

In a formal statement: if the second derivatives of f are all continuous in a neighborhood D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.

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