According to a book that I have an hermitian matrix is definite positive if $X^TA\overline{X}>0$, but here in the forums (and other sources) the definition is given by $X^*AX>0$.
Somehow I'm failing to prove that these two definitions are equivalent. Can someone please explain me this?
This is a problem of mixed up quantifiers.
The one source says (or meant to say, or should say, or did say and the OP miscopied) that a Hermitian matrix $A$ is positive definite if and only if for each nonzero vector $X$, we have the inequality $X^TA\overline X>0$. Similarly, the other source says the condition is that for all nonzero $Y$ you have $Y^* A Y>0$, where I have taken the liberty of renaming variables for clarity.
Now the resolution is clear: the two conditions are the same, if you let the second source's $Y$ equal $\overline X$. If you know the one inequality for all $X\ne0$ you know the other for all $Y\ne0$, and vice versa.