The rank of the sum of outer products distributed according to a Haar measure

Question

While working on the journal version of our paper, we encountered the following problem, which seems to be fairly simple but we could not find the answer:

Suppose $A_i \in \mathbb{R}^{p\times n}$ are i.i.d. Gaussian matrices ($p<n$), and $A_i = V_i \Sigma_i U_i^T$ is the reduced SVD of the matrix. Then taking these $U_i$, we can form $\sum_{i=1}^m U_i U_i^T$. How large $m$ needs to be for this to be full rank?

I think the $U_i$ formed this way follow a Haar measure (meaning its uniformly distributed on Stiefold manifold).

Answer 1

To get a full rank matrix, the size of m should be at least $p$.

Answer 2

For every $i$, $U_i$ is a random pseudo-unitary $(n\times p)$ matrix; then $rank(U_iU_i^T)=p$. Moreover, the $(U_i)_i$ are iid; then the images of the $(U_iU_i^T)_i$ are in general position, that is $rank(\sum_{i=1}^m U_iU_i^T)=\inf(n,mp)$. Thus the required condition is $mp\geq n$.