Let $A$ be an $n\times n$ symmetric matrix. Then, $A$ is a positive semidefinite iff every principal minor of $A$ is $\geq0$; $A$ is a negative semidefinite iff every principal minor of odd order is $\leq0$ and every principal minor of even order is $\geq0$.
Now, let $B$ be a $4\times 4$ symmetric matrix with
- principal minor of odd order $=0$ and principal minor of even order $\geq0$.
- all principal minors $=0.$
I think in both the cases we have positive as well as negative semidefinite matrices, but I think how a single matrix be both +ve and -ve?