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.

# Regression – Difference Between Identity and Diagonal Covariance Matrices

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## Best Answer

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.

(All three have zero covariances between variates)