= Solution
Use the <spectral theorem for real symmetric matrices> to write $\Sigma=Q\Lambda Q^T$, with $Q$ orthogonal and $\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_d)$, $\lambda_i\geq0$. Set $L=Q\Lambda^{1/2}$ and generate
$$
\boxed{X=\mu+LZ,\qquad Z=(X_1,\ldots,X_d)^T\sim N_d(0,I).}
$$
Then $LL^T=\Sigma$. The <characteristic function> is
$$
E e^{it^TX}=e^{it^T\mu}E e^{i(L^Tt)^TZ}
=\exp\left(it^T\mu-\tfrac12t^T\Sigma t\right),
$$
which identifies the <multivariate normal distribution> $N_d(\mu,\Sigma)$. Equivalently, every linear combination is normal, with the required mean and <variance>. A <Cholesky decomposition> can be used for a <positive-definite matrix>. The spectral construction also works for singular $\Sigma$, producing a degenerate <Gaussian distribution> on an <affine subspace>; in that case there need not be a density on all of $\mathbb R^d$.
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