In the case of $X \sim \mathcal N(\mu, \Sigma)$ to sample from it one would need to perform Cholesky decomposition, is it also the case that this is required if $X ~ \sim \mathcal N(\mu, \sigma_{i,j}{I})$
where $\sigma_{i,j}{I}$ denotes a matrix with zeros everywhere except the diagonal, whose entries are not all equal, for example:
$$\begin{bmatrix}.3 & 0 & 0\\0 & .42 & 0\\0 & 0 & .362\end{bmatrix}$$
so that you have a variance matrix instead of a covariance matrix? Would it be appropriate in this case to just sample $X \sim \mathcal N(\mu, \sigma_{ij})$ where $\sigma_{ij}$ is a scalar, the $i,j$ entry of $\sigma_{i,j}{I}$ ?