Is it true that a matrix is always positive semi-definite if it is positive definite?
I think it is not true since I saw a counter-example somewhere.
But the terminology 'semi-definite' and 'definite' seems to imply that 'definite' has strong condition, thus being sufficient condition for semi-definiteness.
So I want to know more detailed mathematical explanation for this proposition.
Assuming we are operating on the real numbers.
Generally, a matrix $M$ is positive definite if $x^TMx >0$ for any non-zero $x \in \mathbb{R}^n$, and positive semidefinite if $x^TMx \ge 0$ for any non-zero $x \in \mathbb{R}^n$.
Therefore, if a matrix is positive definite, it must be positive semi-definite.
UPDATE
You may be confusing the idea that if something is positive semidefinite, that means there is some $x$ for which $x^T Mx = 0$. That does not have to be. Any matrix where $x^T M x > 0$ is both positive semidefinite and positive definite.