Determine if a stationary point of a 3 variable scalar field is a saddle point given its hessian matrix

Question

The Hessian matrix a a stationary point p is : H(p) =$$ \begin{pmatrix} -3 & 0 & 0 \\ 0 & 2 & -4 \\ 0 & -4 & 8 \\ \end{pmatrix} $$ How to determine if it is a saddle point or not. Shall I diagonalize it or is there a faster way to see it ?

Answer 1

Let refer to Sylvester criterion and observe that

• $\det(-3)=-3<0$
• $\det \left(\begin{smallmatrix}-3&0\\0&2\end{smallmatrix}\right)=-6<0$
• $\det(A)=0$

then the signature is $(1,1,1)$.

Answer 2

Let $Q$ be the quadratic form of $H(p)$. Then it is easy to see that

$Q(1,1,0)>0$ and $Q(0,0,-1) <0$. Hence $H(p)$ is indefinite.