How to prove that the principal minor criterion of Hurwitz does not always imply positive semi definiteness if principal minors are $\geq 0$?

Question

Let $ n\ge2$ and $A:=((a_{ij})_{1\leq i,j\leq k}) \in \mathrm{Mat}(n\times n,\mathbb{R} )$ a symmetric matrix with the property that all principal minors are $\geq 0$ or in other words: $\det((a_{ij})_{1\leq i,j\leq k}) \geq 0$ for $k=1,...,n$.

Does it always follow that $A$ is positive semi definite or is there for $n\geq 2$ a matrix $A$ with $ x^{t}Ax<0$ for a $x \in \mathbb{R}^n$?

Answer 1

Consider $A=\left[ {\begin{array}{cc} 0 & 0 \\ 0 & -1 \\ \end{array} } \right]$ which meets the criteria. Now put $x=\left[ {\begin{array}{cc} 0 \\ 1 \\ \end{array} } \right]$ and $x^T Ax=-1$.