We have the matrix $$H=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$$ I want to use the Sylvester's criterion.
The first leading principal minor is equal to $H_1=0$ and the second leading principal minor is equal to $0-1=-1<0$.
We have that not all leading principal minors are positiv so the matrix is not positively definite.
We have that we don't have the sequence $H_1<0$, $H_2>0$ and so the matrix is not negatively definite.
Therefore we either have semidefinite or indefinite, right?
How do we check what holds in this case?