Hessian Matrix

The second-derivative of $f$

Theorem

With $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ as $C^2$. The Hessian of $f$ at $x_0$ is the function:

$$=\frac{1}{2}[h_1,\dots ,h_n]\left[\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial x_1\partial x_1}(x_0)& \dots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}(x_0)\\ ⋮& \dots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}(x_0)& \dots & \frac{{\partial }^{2}f}{\partial x_n\partial x_n}(x_0)\end{array}\right]\left[\begin{array}{c}{h}_1\\ ⋮\\ h_n\end{array}\right]$$

2x2 Example

$$Hf=\left[\begin{array}{cc}\frac{{\partial }^{2}f}{\partial x^2}& \frac{{\partial }^{2}f}{\partial x\partial y}\\ \frac{{\partial }^{2}f}{\partial x\partial y}& \frac{{\partial }^{2}f}{\partial y^2}\end{array}\right]$$