Definition:Hessian Determinant
Definition
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 known as
A Hessian determinant is also known as a Hessian.
However, it is probably better to use the full form so as more easily to distinguish between this and the concept of the Hessian matrix.
Also see
- Results about Hessian determinants can be found here.
Source of Name
This entry was named for Ludwig Otto Hesse.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hessian
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hessian