Condition on minors to be Positive Semidefinite (PSD)

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SUMMARY

For a Hermitian matrix to be Positive Semidefinite (PSD), it is necessary that all principal minors be non-negative. However, it is not required that all minors of order greater than or equal to 2 be non-negative. Specifically, for a 4x4 Hermitian matrix, the minors formed by the elements a_11, a_13, a_31, a_33 and a_22, a_24, a_42, a_44 must be non-negative. The conclusions drawn from the discussion confirm that the first part of the query is negative, while the last part is positive.

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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.
 
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