Hessian Matrices
Text The Hessian matrix H of a function f(x,y,z) is defined as the 3 * 3 matrix with rows [f_{xx}, f_{xy}, f_{xz}], [f_{yx}, f_{yy}, f_{yz}], and [f_{zx}, f_{zy}, f_{zz}].
For twice continuously differentiable functions, a critical point will be a maximum or minimum if and only if the solutions λ to det(H  λI) = 0 are all positive.
Demos
Hessian Matrices
 
This lab displays the graph of a function f(x, y, z) and the graph of the polynomial given by det(H  λI) as a function of λ. H is the Hessian matrix for the point (x_{0}, y_{0}, z_{0}), which you can choose using the hotspots. If the point chosen is a critical point, it will be a maximum if and only if the graph of the polynomial intersects the xaxis for positive values of λ only.

Exercises 1. Are the solutions to det(H  λI) = 0 all positive at the point (0, 0, 0) for f(x, y, z) = x^{2} + y^{2} + z^{2}? Why should this answer be expected (consider the geometric meaning of x^{2} + y^{2} + z^{2})?
2. Are the solutions to det(H  λI) = 0 all positive at the point (0, 0, 0) for f(x, y, z) = x^{2} + y^{2}  z^{2}?
3. Explain why in the two dimensional case, a set of all positive solutions λ to det(H  λI) = 0 implies that f_{xx}f_{yy}  f_{xy}f_{yx} > 0.
