how is noise term defined in linear regression model?

Question

For model $Y=X\beta+ \epsilon$,

where $Y$ is $n\times 1$, $X$ is $n\times p$, $\beta$ is $p\times 1$.

I see two versions of defining $\epsilon$:

e.g. 1. $\epsilon \sim \mathcal{N}(0,\,\sigma^{2}I_{n\times n})$ $\Rightarrow Var(\epsilon) = \sigma^{2}I_{n\times n}$(a matrix)

2. $\epsilon \sim \mathcal{N}(0,\,\sigma^{2})$ $\Rightarrow Var(\epsilon) = \sigma^{2}$(a number)

What is the difference? Thanks in advance.

Answer 1

The first one implies i.i.d noise terms. I.e., as the vector $\epsilon$ is multivariate normal with $0$ pairwise correlations, hence $\epsilon_i$ is independent of $\epsilon_j$ for any $i \neq j$. While the second one says nothing about how $\epsilon_i$ correlates with $\epsilon_j$.