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A condition on principle minors of a symmetric Positive semidefinite (PSD) matrix

  1. Apr 3, 2010 #1
    Hi everyone,
    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?

  2. jcsd
  3. Apr 7, 2010 #2
    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.
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