Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 208 / 4 / c / Solution

Use the spectral theorem for real symmetric matrices to write $\Sigma =Q\Lambda Q^T$, with $Q$ orthogonal and $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_d)$, ${\lambda }_i\ge 0$. Set $L=Q{\Lambda }^{1/2}$ and generateThen $L{L}^T=\Sigma$. The characteristic function iswhich 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 ${\mathbb{R}}^d$.