Hessian
For a scalar valued function \(f\), the Hessian is the matrix of second partial derivatives of the function's result with respect to its arguments.
\[
\mathbf{H}(f) =
\begin{pmatrix}
\frac{\partial^2 f}{\partial x_0^2} & \frac{\partial^2 f}{\partial x_0 \partial x_1} & \dots & \frac{\partial^2 f}{\partial x_0 \partial x_n}\\
\frac{\partial^2 f}{\partial x_1 \partial x_0} & \frac{\partial^2 f}{\partial x_1^2} & \dots & \frac{\partial^2 f}{\partial x_1 \partial x_n}\\
\vdots & \vdots & \ddots & \vdots &\\
\frac{\partial^2 f}{\partial x_n \partial x_0} & \frac{\partial^2 f}{\partial x_n \partial x_1} & \dots & \frac{\partial^2 f}{\partial x_n^2}
\end{pmatrix}
\]