Let's say we have a matrix $X \in \mathbb R^{n\times p}$, where $X_{i,j}$ sampled from a Gaussian $N(\mu, \sigma^2)$, we use $\Phi$ to denote $\{\mu,\sigma\}$ for simplicity.

Now, we sample $m$ different Matrices $X_k$, parametrized by $\Phi_k$ respectively, where $k = 1,\dots m$.

Append them to make a matrix $A\in\mathbb R^{mn\times p}$, calculate $C=AA^T$. When we visualize $C$, we should clearly see the block-diagonal structure.

When we calculate and plot the eigenvalues, we can (mostly) clearly see that there are $m$ significant ones.

Now, do we know anything about the distribution of these eigenvalues (as a function of $\Phi$)?