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$\newcommand\Si\Sigma\newcommand\si\sigma$Consider the spectral decomposition of $\Si$: $$\Si=QDQ^T,$$ where $Q$ is an orthogonal matrix $n\times n$ matrix and $D$ is the diagonal matrix with diagonal entries $\si_1^2>0,\dots,\si_k^2>0,0,\dots,0$. Let $Y:=Q^TX$, so that $X=QY$ and $Y=(Y_1,\dots,Y_k,0,\dots,0)\sim N(0,D)$. So, for any Borel-measurable function $g\colon\mathbb R^n\to\mathbb R$ such that $Eg(X)$ exists, we have $$Eg(X)=Eg((Y_1,\dots,Y_k,0,\dots,0)),$$ and $Y_1,\dots,Y_k$ are independent centered normal random variables with variances $\si_1^2>0,\dots,\si_k^2>0$.