6.21.3 Hessian matrix: hessian

Recall, the Hessian of a function F of n variables x₁,…,x_n is the matrix of second order derivatives:

The hessian command computes the Hessian of a function.

• hessian takes two arguments:
  • expr, an expression involving several variables.
  • vars, a list of the variable names.
• hessian(expr,vars) returns the Hessian of the expression.

Examples.

• Find the Hessian matrix of F(x,y,z)=2x²y−xz³.
  Input:
  hessian(2*x^2*y-x*z^3 , [x,y,z])
  Output:

  ⎡ ⎢ ⎢ ⎣

  4 y 4 x −3 z2 2· 2 x 0 0 −3 z2 0 −2· 3 x z

  ⎤ ⎥ ⎥ ⎦

• To get the Hessian matrix at the critical points:
  Input:
  solve(derive(2*x^2*y-x*z^3,[x,y,z]),[x,y,z])
  Output (the critical points):

  ⎡ ⎣

  ⎡ ⎣

  0,y,0

  ⎤ ⎦

  ⎤ ⎦

  Input (to evaluate the Hessian at these points):
  subst([[4*y,4*x,-(3*z^2)],[2*2*x,0,0], [-(3*z^2),0,6*x*z]],[x,y,z],[0,y,0])
  Output:

  ⎡ ⎢ ⎢ ⎣

  4 y 0 0 0 0 0 0 0 0

  ⎤ ⎥ ⎥ ⎦