I have a question regarding the positive semidefiniteness (PSD) of a symmetric matrix. I have seen different conditions stated in papers and journals, and I end wondering which are correct. Three definitions I have seen: 1) A symmetric matrix is PSD if and only if all its principal minors are non- negative. 2) A symmetric matrix is PSD if and only if the sum of the principal minors of order k (for each k) are non-negative. 3) A symmetric matrix is PSD if and only if the leading principal minors are non-negative.
Those conditions do not seem equivalent to me (1 => 2 but not the converse for example. Can somebody provide guidance / references? I have found the above definitions in reputable books or journals.