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I heard that

for a Hermitian or symmetric matrix, if it is positive semi-definite, then all its square submatrices (not just those along diagonal) are non-negative

  1. I suspect the statement is wrong.

    There can be different interpretations of "non-negative".

    If the statement meant to say "all its square submatrices (not just those along diagonal) have non-negative determinants", then matrix $[2,-1;-1,2]$ would be a positive semidefinite matrix but the square submatrix $[-1]$ doesn't have nonnegative determinant.

    If the statement meant to say "all its square submatrices (not just those along diagonal) are positive semi-definite", then matrix $[2,-1;-1,2]$ would still be a positive semidefinite matrix but the square submatrix $[-1]$ isn't positive semi-definite.

    So I wonder what the statement might actually want to mean? What is the closest correct statement to it?

  2. Is the converse of the statement also true, i.e. "for a Hermitian or symmetric matrix, if all its square submatrices (not just those along diagonal) are non-negative, then it is positive semi-definite"?

Thanks!

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