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 $\mathbb{A}_{(n-1)\times p}$?


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