First let me check my definition is correct:
- Let $A\in M_n(\Bbb C)$. If $(\forall x\in\Bbb C^n\setminus\{0\},~x^*Ax>0)$, then we called $A$ positive definite.
- Let $A\in M_n(\Bbb R)$. If $(\forall x\in\Bbb R^n\setminus\{0\},~x^TAx>0)$, then we called $A$ positive definite.
First question: is my definition stated above "correct" (namely, standard)?
Second, now, if $A=\begin{bmatrix}2&3\\ 1&4\end{bmatrix}$. By the second meaning/definition of positive definite, it really is positive definite. However, how about in view of the first definition? $A$ is not Hermitian, hence it is impossible that $(\forall x\in\Bbb C^n\setminus\{0\},~x^*Ax>0)$. Then shouldn't $A$ be non positive definite?
Third, if $A\in M_n(\Bbb R)$ is real, symmetric. Then does "$\forall x\in\Bbb R^n\setminus\{0\},~x^TAx>0$" necessarily implies "$\forall x\in\Bbb C^n\setminus\{0\},~x^*Ax>0$"?
A positive definite matrix is a Hermitian matrix which satisfies the condition that you described (the first one). Therefore, your matrix $A$ is not definite positive. However, note that where you wrote $x\in\mathbb{C}\setminus\{0\}$, you should have written $x\in\mathbb{C}^n\setminus\{0\}$.