Is a Zero Principal Minor in PSD Matrices Indicative of Smaller Zero Minors?

[NaturePaper]

Hi everyone,
Let A=(a_{ij}) be a symmetric (i.e., over reals) PSD matrix. Then is the following correct?

"If any principle minor ( \ne A ) be zero, then all principle minor contained in this minor should also be zero".

I can not prove or disprove it..any help?

By the way how the result will change if we consider Hermitian matrix (over complex) instead of symmetric matrix?

Thanks

 

[NaturePaper]

Oh..I got the answer. Its not correct. Consider the diagonal matrix: D={1,1,0,1,...}. Clearly $A_33=0$ but $A_22$ is non-zero.