Condition on minors to be Positive Semidefinite (PSD)

[NaturePaper]

Hi everyone,

I know that for a hermitian matrix to be PSD it is necessary that every principal minor [i.e, the minors obtained by deleting all the last i rows and columns for all i=(n-1)(-1)0].

I want to know if it is necessary that all minors of order>=2 be non-negative.

Particularly, for the 4x4 hermitian matrix

a_11 a_12 a_13 a_14
a_21 a_22 a_23 a_24 (please read it as a matrix)
a_31 a_32 a_33 a_34
a_41 a_42 a_43 a_44

is it necessary that the minors
a_11 a_13
a_31 a_33

and

a_22 a_24
a_42 a_44

should be non-negative?

Please help.

 

[NaturePaper]

The answer to this problem can be found https://www.physicsforums.com/showthread.php?t=316436".

In short, the answer to the first part is
negative and the answer to the last part is positive.