# Difference between identity and diagonal covariance matrices

thanks in advance for the help.

Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the difference between a full covariance matrix and a diagonal covariance matrix is that there is no correlation between predictors with the diagonal matrix. I'm not quite sure of the differences between the identity and diagonal matrices though.

An identity covariance matrix, $\Sigma=I$ has variance = 1 for all variables.
A covariance matrix of the form, $\Sigma=\sigma^2I$ has variance = $\sigma^2$ for all variables.
A diagonal covariance matrix has variance $\sigma^2_i$ for the $i^\text{th}$ variable.