I am working with a standard multivariate linear regression model ($Y = X \lambda + \epsilon$, $Y$ and $\epsilon$ of length $n$, $X$ is $n$ x $m$, $\lambda$ of length $m$). $X$ is said to be "column-standardized", which I take to mean each column has mean $0$ and variance $1$, and each $\epsilon_i \sim N(0, \sigma^2)$. I am looking for the distribution of $X^T \epsilon$. Specifically, I have two questions.
If $X$ is column-standardized, does that mean each $X_{ij} \sim N(0, 1)$? It seems like it should, since this is just a z-transformation, but then I'm curious as to why the procedure needs to be done on columns separately. Is it because we assume that each column comes from a unique distribution with different original variances?
If every column in standardized, then each entry $(X^T\epsilon)_j = \sum_{i=1}^{n} X_{ij} \epsilon_i$, which I think is a linear combination of standard normal distributions where the coefficients are also normal. Sums of normals are normal, and so I think that $(X^T\epsilon)_j \sim N(0, \_)$, but I'm having a hard time figuring out the variance. I don't think it's $n\sigma^2$, but I can't figure out what else it would be.
Any help would be greatly appreciated!