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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?

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For each integer $k$ one has $H_kx=x-\bar x 1_k$ and the matrix $H_k$ is symmetric.

Thus the $j$-th colmun of $H_p\mu$ is $\mu_{:,j} - \bar\mu_{:,j} 1_p$ where $\mu_{:,j}$ is the $j$-th column of $\mu$. Similarly the $(j,i)$-th entry of $H_n{(H_p\mu)}'$ is $\mu_{ij}-\bar\mu_{:,j} - \bar \mu$ and. Then transpose since $H_p\mu H_n = {\bigl(H_n{(H_p\mu)}'\bigr)}'$.

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