# Semi positive definite matrix is always positive definite matrix

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

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.
• 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. – Navdeep May 27 '17 at 4:03
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 $z^TAz>0$