Consider a random sample $X = [X_1 \;X_2\; \cdots\; X_n]\in \mathbb{R}^{p\times n}$ from the multivariate normal distribution, where the column vectors $X_i$ are independently and identically distributed as $N_p(\mu,\Sigma)$. Let $\bar{X} = (1/n)\sum_{i=1}^n X_i$ and $S = 1/(n-1) \sum_{i=1}^n (X_i-\bar{X})(X_i-\bar{X})^\top$.
Show that the sample covariance matrix ($S$) is unbiased estimator of $\Sigma$.
Show that $(n-1)S$ is distributed as the Wishart random variable with parameters $n -1$ and $\Sigma$ , that is $(n − 1)S \sim W_p(n − 1, \Sigma)$
Attempt: I have managed myself to find the first demonstration. I need some insight for the second.