Sampling Multivariate Normals via Spectral Decomposition

You need to generate samples from a multivariate normal distribution $X \sim N(0, \Sigma)$ where $\Sigma$ is an $N \times N$ positive semi-definite covariance matrix. The standard approach is Cholesky decomposition, but suppose Cholesky is off the table -- maybe $\Sigma$ is singular or nearly singular, or you just want to understand the alternative. 1. Explain how to use the spectral (eigenvalue) decomposition of $\Sigma$ to generate these samples. Give the step-by-step procedure. 2. Prove that your method produces the correct covariance structure, i.e., that $\text{Cov}(X) = \Sigma$. 3. When would you prefer spectral decomposition over Cholesky in practice?