Category:Hessians

This category contains results about Hessians.
Definitions specific to this category can be found in Definitions/Hessians.

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}$