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Condition on minors to be Positive Semidefinite (PSD)

  1. Jan 9, 2009 #1
    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


    a_22 a_24
    a_42 a_44

    should be non-negative?

    Please help.
    Last edited: Jan 9, 2009
  2. jcsd
  3. Jun 27, 2009 #2
    The answer to this problem can be found here.

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