Concept

[𝐻𝑓(𝑥)]𝑖𝑗=𝜕2𝑓(𝑥)𝜕𝑥𝑖𝜕𝑥𝑗

which is symmetric.

property

At critical point, 𝐻𝑓(𝑥) is

  • positive definite 𝑥 is minimum
  • negative definite 𝑥 is maximum
  • indefinite 𝑥 is saddle point
  • singular, then various pathological situations are possible

Example

𝑓=1+(𝑥1)2+(𝑥4)2

Hessian matrix is 1×1 symmetric matrix.