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I've found this page for calculating the minors (Hauptminoren) of the Hessian matrix to determine which of the critical points of the matrix correspond to a maximum, a minimum or a saddle point. They say the matrix is positive definite if all $|q_A| > 0$ and negative if $|q_A|(-1)^k>0$, and a saddle point if otherwise. But what happens if one of $|q_A|$ is zero? No conclusion possible?

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Possibly a related post:… – Martin Sleziak Jun 27 '11 at 12:47
Yeah, I'd say related. – Reactormonk Jun 28 '11 at 15:14
up vote 1 down vote accepted

Classical example is the function

$$F: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto x^4+y^4$$

which has a minimum at $(0,0)$ but the Hessian matrix is the zero matrix in that point.

On the other hand,

$$F: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto x^3+y^3$$

also has Hessian matrix equal to the zero matrix in $(0,0)$, but this time you don't have a minimum or maximum.

So, the Hessian alone doesn't allow you to conclude in these cases.

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