I'm trying to understand the expression $Cov(\hat y,\hat \epsilon)$ in regards to the usual linear regression model/assumptions $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... \beta_n x_n +\epsilon$. Also, $x_i$ are not considered to be random variables. Rather, they are considered to be fixed measurements with negligible error.
Each $\hat y_i$ is the sum of the $x_i$s weighted by the $\hat \beta_i$s (with $x_0=1)$. And each $\hat \epsilon_i$ is the difference ($y_i - \hat y_i$).
Are $\hat y,\hat \epsilon$, and for that matter, $\hat \beta,$ random vectors?
If u,v are random vectors, then $Cov(u,v)$ is the matrix of elements $Cov(u_i,v_j)$
If $u,v$ are not random vectors, then $Cov(u,v)$ is the scalar $\Sigma u_i v_i$.