Covariance matrices for jointly gaussian distributions

Question

Is the 2nd parameter of a jointly gaussian distribution, the covariance matrix $\Sigma$, always just a diagonal matrix with values of the variances as the diagonal entries?

When I google image covariance matrix most of them just appear like this, but a couple have nonzero entries in indices other than the diagonals.

Answer 1

$\Sigma$ is properly called variance-covariance matrix, has variances on the diagonal and covariances off diagonal. It is diagonal if the random variables involved have zero covariances. Would seeing plots help? Try here or here.