In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. Hessian Matrices are often used in optimization problems within Newton-Raphson's method.
Example 1: Computing a Hessian
Problem: Compute the Hessian of .
First compute both partial derivatives:
With these, we compute all four second partial derivatives:
The Hessian matrix in this case is a $ 2\times 2$ matrix with these functions as entries:
Problem: the function , where is a matrix, is a vector of length and is a constant.
- Determine the gradient of : .
- Determine the Hessian of : .
- compute the gradient :
- compute the Hessian :