Hessian (mathematics)

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".

Definitions and properties

Suppose $f : \R^n \to \R$ is a function taking as input a vector $\mathbf \in \R^n$ and outputting a scalar $f(\mathbf) \in \R.$ If all second-order partial derivatives of $f$ exist, then the Hessian matrix $\mathbf$ of $f$ is a square $n \times n$ matrix, usually defined and arranged as follows:
\mathbf H_f= \begin
\dfrac & \dfrac & \cdots & \dfrac \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \vdots & \vdots & \ddots & \vdots \\ .2ex \dfrac & \dfrac & \cdots & \dfrac
\end,
or, by stating an equation for the coefficients using indices i and j,
$(\mathbf H_f)_ = \frac.$

If furthermore the second partial derivatives are all continuous, the Hessian matrix is a symmetric matrix by the symmetry of second derivatives.

The determinant of the Hessian matrix is called the .

The Hessian matrix of a function $f$ is the Jacobian matrix of the gradient of the function $f$; that is: $\mathbf(f(\mathbf)) = \mathbf(\nabla f(\mathbf)).$

Applications

Inflection points

If $f$ is a homogeneous polynomial in three variables, the equation $f = 0$ is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most $9$ inflection points, since the Hessian determinant is a polynomial of degree $3.$

Second-derivative test

The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point $x$ is a local maximum, local minimum, or a saddle point, as follows:

If the Hessian is positive-definite at $x,$ then $f$ attains an isolated local minimum at $x.$ If the Hessian is negative-definite at $x,$ then $f$ attains an isolated local maximum at $x.$ If the Hessian has both positive and negative eigenvalues, then $x$ is a saddle point for $f.$ Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite.

For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of Morse theory.

The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then $x$ is a local minimum, and if it is negative, then $x$ is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is zero, then the second-derivative test is inconclusive.

Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero. Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with the $1 \times 1$ minor being negative.

Critical points

If the gradient (the vector of the partial derivatives) of a function $f$ is zero at some point $\mathbf,$ then $f$ has a (or ) at $\mathbf.$ The determinant of the Hessian at $\mathbf$ is called, in some contexts, a discriminant. If this determinant is zero then $\mathbf$ is called a of $f,$ or a of $f.$ Otherwise it is non-degenerate, and called a of $f.$

The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points.

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 eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature. (See .)

Use in optimization

Hessian matrices are used in large-scale optimization