Eigenvalue distribution of low rank matrices

Question

Is there any relation between the eigenvalue distribution of $H H^\dagger$ and $\sum_{i=1}^{r}\alpha_i h_i h_i^\dagger$? where H is a $n\times r$ i.i.d complex Gaussian matrix ($r<<n$) and $h_i$ is a $n\times 1$ i.i.d complex Gaussian vector. $\dagger$ is Hermitian operator.

Answer 1

This is a special case of the more general problem solved in On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices (2015). If all the $\alpha_i$'s are the same, the two distributions are identical upon rescaling, if they differ, the relation is more complicated, as worked out in this paper.