Definition:Hessian
Definition
Hessian Matrix
Let $\mathbf f: \R^n \to \R$ be a real-valued function on $n$ independent variables.
The Hessian matrix of $\mathbf f$ is the square matrix of order $n$ containing the second partial derivatives of $\mathbf f$:
- $\begin {pmatrix} \dfrac {\partial^2} {\partial x_1 \partial x_1} & \dfrac {\partial^2} {\partial x_1 \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_1 \partial x_n} \\
\dfrac {\partial^2} {\partial x_2 \partial x_1} & \dfrac {\partial^2} {\partial x_2 \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac {\partial^2} {\partial x_n \partial x_1} & \dfrac {\partial^2} {\partial x_n \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_n \partial x_n} \\ \end {pmatrix}$
That is, the $\tuple {i, j}$th element contains $\dfrac {\partial^2} {\partial x_i \partial x_j}$.
Hessian Determinant
Let $\mathbf f: \R^n \to \R$ be a real-valued function on $n$ independent variables.
The Hessian determinant of $\mathbf f$ is the determinant of the Hessian matrix of $\mathbf f$:
- $\begin {vmatrix} \dfrac {\partial^2} {\partial x_1 \partial x_1} & \dfrac {\partial^2} {\partial x_1 \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_1 \partial x_n} \\
\dfrac {\partial^2} {\partial x_2 \partial x_1} & \dfrac {\partial^2} {\partial x_2 \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac {\partial^2} {\partial x_n \partial x_1} & \dfrac {\partial^2} {\partial x_n \partial x_2} & \cdots & \dfrac {\partial^2} {\partial x_n \partial x_n} \\ \end {vmatrix}$
Also see
- Results about Hessians can be found here.
Source of Name
This entry was named for Ludwig Otto Hesse.