Consider the matrix $A$ defined as follows $\begin{pmatrix}a & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$, where $a > 0$.
According to the definition, $A$ is positive definite if for any $x \in \mathbb{R}^{3} \backslash\{0\}$, $x^{T}Ax > 0$. Calculating $x^{T}Ax = x_{1}^{2}a$ which is clearly bigger than zero. Hence the matrix must be positive-definite.
From the other hand, it has one positive and two zero eigen values which suggests that the matrix is not positive-definite but only positive semi-definite.
Wolfram alpha also says that this matrix is not positive-definite but positive semi-definite. What am I missing?