What is the rank of a normally distributed matrix that is multiplied by rank-r projection matrices from left and right.

Question

Let $Z \in \mathbb{R}^{m_1\times m_2}$ be a full rank matrix such that $Z_{ij} \sim \mathcal{N}(0,1)$. Moreover let $U \in \mathbb{R}^{m_1\times R}$ be a matrix with R orthonormal columns such that $U^TU = I$, and let $V \in \mathbb{R}^{m_2\times R}$ be a matrix with R orthonormal columns such that $V^TV = I$, where $m_1,m_2 \geq R$. Then, define two rank-R orthogonal projection matrices $P_u = UU^T$ and $P_v = VV^T$. The article I am reading states that it is easy to show that matrix $X = P_uZP_v$ has rank R almost surely. Unfortunately, I was unable to prove this statement and would appreciate any help. Thank you.

Answer 1

Consider the case where $m_1=m_2$ is even and $R=\frac{m_1}{2}$. From here I select select $Z:= I$ and orthogonal matrix $Q\in \mathbb R^{m_1\times m_1}$

$Q:=\bigg[\begin{array}{c|c|c|c|c} U &V\end{array}\bigg]$

Then
$X = P_uZP_v=\mathbf 0$ and
$R\gt \text{rank}\big(X)= 0$
which contradicts your claim.