Hessian matrix

Concept

$$[H_f(x){]}_{ij}=\frac{{\partial }^{2}f(x)}{\partial x_i\partial x_j}$$

which is symmetric.

property

At critical point, $H_f(x^{\ast })$ is

• positive definite $⟹$ $x^{\ast }$ is minimum
• negative definite $⟹$ $x^{\ast }$ is maximum
• indefinite $⟹$ $x^{\ast }$ is saddle point
• singular, then various pathological situations are possible

Example

$$f=1+(x-1{)}^2+(x-4{)}^2$$

Hessian matrix is $1\times 1$ symmetric matrix.