If we have a sample with size $n$ $(\mathbf x_1,\dots, \mathbf x_n)$, where $\mathbf x_j \in \mathbb R^p$, the scatter matrix is defined as $$ S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})(\mathbf {x} _{j}-{\overline {\mathbf {x} }})^{T},$$ where $\overline{\mathbf {x}}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}$ is sample mean. I found on wiki page that:
$S$ is positive definite if there exists a subset of the data consisting of $p$ linearly independent observations...
but I don't know how to show that (I can show only that it is positive semi-definite).