# Understanding covariance of errors in regression

I am having a hard time understanding the elements of an error covariance matrix for a class. Can someone clarify?

First, the diagonal. The variance is $E(e_i^2) - E(e_i)^2$. $E(e_i) = 0$, so it's just $E(e_i^2)$. Ok . . . so that is the square of the actual value of $e_i$? $y_i - ŷ_i$? A non-zero number, i.e., how much that error value "varies" from the expected zero? So, how does the E work in this?

The off-diagonal terms = $E(e_ie_j)$? The expected value of their product? What is that telling you?

My notes say: "The graph suggests that the variance around the regression line is constant: $\sigma^2 = \sigma_i^2$ "

I can't see it. Where is the constant? The ei values are all over the place, though their mean/expected value does look like zero. And why would the off-diagonal terms be zero? I would expect constant error variance to look something like: