 
 
 
13.7.3  Hessian matrix
Recall, the Hessian of a function F of n variables
x1,…,xn 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.
| hessian(2*x^2*y-x*z^3,[x,y,z]) | 
|  | | |  | | ⎡ ⎢
 ⎢
 ⎣
 | | 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:
| solve(derive(2*x^2*y-x*z^3,[x,y,z]),[x,y,z]) | 
Output (the critical points):
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]) | 
 
 
