Hessian Matrix

The Hessian Matrix is a square matrix consisting of second-order partial derivatives of a function. It is primarily used in the second derivative test, but has uses in optimisation.

Definition §

Hessian Matrix (Bivariate) ^definition-bivariate

Let $f(x,y)$ be a twice-differentiable bivariate function. Then the Hessian matrix for $f$ is defined to be:

$${\mathbf{H}}_f=\left[\begin{array}{cc}\frac{{\partial }^{2}f}{\partial x^2}& \frac{{\partial }^{2}f}{\partial x\partial y}\\ \frac{{\partial }^{2}f}{\partial y\partial x}& \frac{{\partial }^{2}f}{\partial y^2}\end{array}\right]=\left[\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{yx}& f_{yy}\end{array}\right]$$

If $f$ is of second-order smoothness $f\in C^2$, then by the smoothness theorem for mixed partial derivatives, ${\mathbf{H}}_f$ is a symmetric matrix

$${\mathbf{H}}_f=\left[\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{xy}& f_{yy}\end{array}\right]$$

The determinant of the Hessian matrix is known simply as the Hessian, $\det (\mathbf{H})$

Hessian Matrix (Multivariate) ^definition-multivariate

Let $f(x_1,\dots ,x_n)$ be a twice-differentiable function. Then the Hessian matrix for $f$ is defined to be:

$${\mathbf{H}}_f=\left[\begin{array}{cccc}\frac{{\partial }^{2}f}{\partial x_1^2}& \frac{{\partial }^{2}f}{\partial x_1\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ \frac{{\partial }^{2}f}{\partial x_2\partial x_1}& \frac{{\partial }^{2}f}{\partial x_2^2}& \cdots & \frac{{\partial }^{2}f}{\partial x_2\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \frac{{\partial }^{2}f}{\partial x_n\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_n^2}\end{array}\right]$$

Applications §

• Used in Taylor Series