Hessian matrix of a function $f:\mathbb{R}^n\rightarrow\mathbb{R}\;$ is used to determine if the critical point of the function is maxima, minima or saddle point.
What if after deriving the function $\;f\;$ twice, we receive a hessian matrix, that is constant (not dependent on the variables $x_1,…,x_n$)? Does it become useless in this case for determining types of critical points?
If so, how can I figure out, if the point is a local maxima or minima, if the hessian matrix is constant?
Best Answer
Remember the value of the Hessian matrix at the critical point tell us the type of point it is, i.e. a minimum, maximum or saddle. It matters not what the Hessian is doing away from these points.
The fact your Hessian matrix is constant indicates the function is a quadratic so it can have at most one such point.
If the (constant) Hessian matrix is positive definite then the point is a minimum. If $H$ is negative definite then it's a maximum and if $H$ is indefinite then the point is a saddle point.
Imagine $f(x,y) = x^2-y^2$ then there is a critical point at $(0,0)$ but if we move from this point in the $y$ direction clearly we get a negative number $f(0,y) = -y^2$ but if we move in the $x$ direction we get a positive number $f(x,0)=x^2$.
The Hessian matrix in this case is
$$H=\left[\begin{matrix}2 & 0\\\ 0 & -2\end{matrix}\right]$$
which is indefinite, indicating $(0,0)$ is a saddle point.