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Can we say that a semi-positive definite matrix is always a positive definite matrix but not vice versa? If No, kindly explain.

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No; in fact, the opposite holds. If a symmetric matrix $A$ is positive definite, then $x^TAx > 0$ for all nonzero $x$. If $x=0$, then $x^TAx = 0$, and so in general $x^TAx \geq 0$, and so $A$ is positive semi-definite.

Conversely, note that the zero matrix is positive semi-definite but not positive definite.

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  • $\begingroup$ Please refere to (en.wikipedia.org/wiki/Positive-definite_matrix). It says that for positive semi-definite, 0 is allowed but not for positive definite matrix. It means $$ x^TAx>0 $$ for positive definite matrix and $$ x^TAx>=0 $$ for semi-positive definite matrix. According to it, positive definite matrix is subset of positive definite matrix. $\endgroup$ – Navdeep May 27 '17 at 4:03
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    $\begingroup$ Right, that's why positive semi-definite matrices aren't always positive definite. $\endgroup$ – florence May 27 '17 at 4:05
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Any positive definite matrix is also positive semi-definite. This is not true in the other order. This is because if A is positive definite, then for any non-zero vector $z$ we have $z^TAz>0$ this implies $z^TAz\ge0$ which is the definition of a positive semi-definite matrix. You can see that the other way is not true because if $z^TAz=0$ then $z^TAz\ge0$ but not $z^TAz>0$

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