Hessian matrix

Named after Otto Hesse Hessian matrix is a matrix which is an analogue of the second derivative of a function in the multidimensional real analysis.

The Hessian matrix arises in the approximation of a multi-dimensional function in the Taylor expansion. It is. Including in connection with the optimization of systems of meaning that are described by several parameters, such as often occur, for example in economics, physics, theoretical chemistry or engineering


Let be a twice continuously differentiable function. Then the Hessian matrix of the point is defined by

With the second partial derivatives are known. The Hessian matrix corresponds to the transpose of the Jacobian matrix of the gradient, but with continuous second derivatives because of the interchangeability of Differentiationsreihenfolge symmetrical so that the transpose of the matrix does not change.


  • For, applies and so
  • The function, which assigns to each vector in its Euclidean norm is, for all twice continuously differentiable and it is by the chain rule


Taylor expansion

The Taylor expansion of a twice continuously differentiable function with a development agency begins with

The second order of the expansion terms are thus given by the quadratic form whose matrix is evaluated at the development site Hessian matrix.

Extreme values

With the help of the Hessian matrix is the essence of the critical points of a figure in determine. To do this determines the definiteness of the Hessian matrix for the previously determined critical points.

  • The matrix is ​​positive definite in a position as is located at this point, a local minimum of the function.
  • Is there definite Hessian matrix is negative, it is a local maximum.
  • If it is indefinite, then it is a saddle point of the function.

If the Hessian matrix at the site under investigation is only semi-definite, so fails this criterion and the character of the critical point must be determined by other means. Which of these situations exists, may - be decided, for example, using the sign of the eigenvalues ​​of the matrix or of its principal minors - as described in definiteness.

Example: The function has definite at a critical point, but is neither positive nor negative. The function has no extremum at this point, but a saddle point by intersecting two contour lines.


There is also a correlation between the positive definiteness of the Hessian matrix and the convexity of a twice continuously differentiable function defined on an open, convex set: such a function is precisely then convex if its Hessian matrix is positive semidefinite everywhere in. If the Hessian matrix is positive definite even in, then the function is convex on strictly. Accordingly, the following applies: A twice continuously differentiable function is convex on its definition exact amount then concave if its Hessian matrix is negative semidefinite. If the Hessian matrix even negative definite, so is a strictly concave.

Is on its definition amount strictly convex, so at the most, to a global minimum. Each local minimum is also the ( only ) global minimum. Is strictly concave, so at most has a global maximum. Each local maximum is also her ( single ) global maximum.