Suppose we have $X_{p\times n} \sim N_p(\mu, \Delta \otimes \Sigma)$. Then what is the distribution of $H_p X H_n$ where $H_p = I_p - 1_p1_p^T/p$ and $H_n = I_n - 1_n 1_n^T/n$?
I know that it should be $H_p X H_n \sim N_p(H_p \mu H_n, (H_n^T \Delta H_n) \otimes (H_p^T \Sigma H_p))$, but what are the entries of $H_p \mu H_n$ and $H_n^T \Delta H_n$? Are there any nice clean properties of the centering matrix that can help?