# Linear least squares decomposition of a submatrix of a gaussian random matrix

Suppose $X$ is an $n\times p$ random matrix whose rows are picked iid according to $N(0,\Sigma)$, $\Sigma$ being $p\times p$ symmetric, positive definite. Let $S,T\subset \{1,2,\ldots,p\}$ be such that $|S|=|T|=k$, $S\neq T$. Let $X_S$ denote the $n\times k$ matrix comprising of the $k$ columns of $X$ indexed by $S$. Similar definition for $X_T$, and $X_{S\backslash T}$. Let $\Sigma_{UV}$ denote the submatrix of $\Sigma$ with rows indexed by $U\subset\{1,\ldots,p\}$, and columns indexed by $V\subset\{1,\ldots,p\}$.

Can anybody please prove that conditioned on $X_T$, $X_{S\backslash T}$ can be decomposed as \begin{equation*} X_{S\backslash T}=X_T(\Sigma_{TT})^{-1}\Sigma_{T(S\backslash T)} + E_{S\backslash T} \end{equation*} where, $E_{S\backslash T}\in\mathbb{R}^{n\times|S\backslash T|}$ is a random matrix independent of $X_T$ with rows drawn iid from $N(0,\Gamma(T,S))$, with $\Gamma(T,S)\triangleq \Sigma_{S\backslash T(S\backslash T)}-\Sigma_{(S\backslash T) T}(\Sigma_{TT})^{-1}\Sigma_{T(S\backslash T)}$?

Is this some kind of a standard result? It looks like the linear least squares decomposition for vectors, but I am not able to visualize this for a matrix. Any pointer would be highly appreciated. Thanks in advance :-)

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