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

"If any principle minor ( [tex] \ne A [/tex] ) 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