Suppose $\Sigma^{XY} \in R^{d \times d}$ is the empirical cross-covariance matrix computed from samples $X \in R^{n \times d}$ and $Y \in R^{n \times d}$. Suppose $Z$ is the result of applying a random permutation $\Pi$ on $Y$’s rows, and that $\Sigma^{XZ}$ is the empirical cross covariance matrix between $X$ and $Z$. Is there a relation between $\Sigma^{XY}$ and $\Sigma^{XZ}$?
Thanks.