Hessian matrix

Encyclopedia : H : HE : HES : Hessian matrix

In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. Given the real-valued function

f(x₁, x₂, ..., x_n),

if all partial second derivatives of f exist, then the Hessian matrix of f is the matrix

H(f)_ij(x) = D_i D_j f(x)

where x = (x₁, x₂, ..., x_n). That is,

[H(f) = \begin\frac & \frac & \cdots & \frac \\\frac & \frac & \cdots & \frac \\\vdots & \vdots & \ddots & \vdots \\\frac & \frac & \cdots & \frac\end]

The term Hessian was coined by James Joseph Sylvester, named for German mathematician Ludwig Otto Hesse, who had used the term functional determinants. Hessian matrices are used in large-scale optimization problems within the BFGS method.

Contents

1 Mixed derivatives and symmetry of the Hessian
2 Critical points and discriminant
3 Second derivative test
4 Bordered Hessian
5 Vector-valued functions
6 See also

Mixed derivatives and symmetry of the Hessian

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. For example,

[\frac \left( \frac \right) = \frac \left( \frac \right)]

This can also be written as:

[f_ = f_ \,]

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

Critical points and discriminant

If the gradient of f (i.e. its derivative in the vector sense) is zero at some point x, then f has a critical point at x. The determinant of the Hessian at x is then called the discriminant. If this determinant is zero then x is called a degenerate critical point of f. Otherwise it is not degenerate.

Second derivative test

The following test can be applied at a non-degenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive.

Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive. However, more can be said from the point of view of Morse theory.

In view of what has just been said, the second derivative test for functions of one and two variables are simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, if it is negative then x is a local maximum; if it is zero then the test is inconclusive. In two variables, the discriminant 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.

Bordered Hessian

A Bordered Hessian is utilized for the second derivate test in certain constrained optimization problems. Given the function as before:

f(x₁, x₂, ..., x_n),

but adding a constraint function such that:

g(x₁, x₂, ..., x_n) = c,

the bordered hessian appears as:

[H(f,g) = \begin0 & \frac & \frac & \cdots & \frac \\\frac & \frac & \frac & \cdots & \frac \\\frac & \frac & \frac & \cdots & \frac \\\vdots & \vdots & \vdots & \ddots & \vdots \\\frac & \frac & \frac & \cdots & \frac\end]

Then the same rules of positive definite and negative definite apply as before.

Vector-valued functions

If f is instead vector-valued, i.e.

f=(f₁, ..., f_n),