# What will be the distribution of the following quantity?

Suppose $$X_1,X_2,\dots,X_n$$ are p-dimensional random variables with distribution $$N_p(\mu,\Sigma)$$. Let $$X_{n\times p}$$ is the data matrix. $$\mathbb{Z}_{n\times p}= (n-1)^{-1/2}\left(I_n - n^{-1}\mathbb{1}\mathbb{1}^t\right)XD^{-1/2}_S$$. $$D_s$$ is the diagonal of the sample covariance matrix. Let there is an orthogonal matrix $$\mathbb{O}_{n\times n}=[(1/\sqrt{n})\mathbb{1},\mathbb{O}_{2:n}]$$ where columns of $$\mathbb{O}_{2:n}$$ are obtained by Gram-Schmidt orthogonalization. $$\begin{bmatrix} \mathbb{0}^t\\ \mathbb{A} \end{bmatrix} = \mathbb{O}^t\mathbb{Z}$$

What will be the distribution of columns of $$\mathbb{A}_{(n-1)\times p}$$?