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Does it has something to do with the determinant of A? I saw two seperate websites - one which claims that when the determinant of A is zero, and the other claims that when the determinant of A is not zero, then the positive semi-definite matrix is positive definite.

Can someone explain to me which is the correct answer, and the reason behind it? Thank you.

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take a look: en.wikipedia.org/wiki/Sylvester's_criterion –  Ilya Oct 12 '11 at 20:50
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up vote 4 down vote accepted

If the determinant of a positive semi-definite matrix is not zero (i.e., the matrix is non-singular), then it is positive definite. Because, determinant=Product of eigenvalues and a positive definite matrix have all positive (strictly greater than zero) eigenvalues.

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Yeah. PSD means the eigenvalues are all >= 0. PD means the eigenvalues are all > 0. The determinant is the product of the eigenvalues. Best way to think of the difference between PD and SPD IMO. –  Jay Lemmon Oct 12 '11 at 20:52
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