Let's say I have a $p$-dimensional multivariate Gaussian distribution. And I take $n$ observations (each of them a $p$-vector) from this distribution and calculate the sample covariance matrix $S$. In this [paper](http://amstat.tandfonline.com/doi/full/10.1198/jasa.2011.tm10155), the authors state that the sample covariance matrix calculated with $p > n$ is singular. 

* How is it true or derived?
* Any explanations?