Chapter 7: Symmetric Matrix. In the chapter mind map this icon labels Cholesky App: Multivariate Gaussian Sampling. The discussion below is excerpted and lightly edited from § Example: Multivariate Gaussian Generation in Mathematics for AI and Machine Learning. Related material also appears in Chapter 7 (Cholesky $A = LL^\top$).

Generate $\mathbf x \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$, where $\boldsymbol\mu \in \mathbb{R}^{N \times 1}$ and $\Sigma \in \mathbb{R}^{N \times N}$ is positive definite. (See the matrix chapter for the definition of the multivariate Gaussian distribution.)

Since $\mathbf{z} \sim \mathcal{N}(0, I)$, we have $\mathbb{E}[\mathbf z] = 0$ and $\mathrm{Cov}(\mathbf z) = I$. The linear transformation $\mathbf y = L\mathbf z$ gives:

• $\mathbb{E}[\mathbf y] = L \cdot 0 = 0$
• $\mathrm{Cov}(\mathbf y) = L \cdot I \cdot L^\top = LL^\top = \Sigma$

Therefore, $\mathbf y = L\mathbf z \sim \mathcal{N}(0, \Sigma)$. Adding the mean shift, we obtain: $\mathbf x = \boldsymbol\mu + \mathbf y \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Therefore, $\mathbf x$ can be generated in the following steps:

1. Generate $\mathbf z \sim \mathcal{N}(0, I)$. 2. Do Cholesky decomposition $\Sigma = LL^\top$. 3. Set $\mathbf x = \boldsymbol\mu + L\mathbf z$.