When will PCA be equivalent to ICA?

$X = AS$ where $A$ is my mixing matrix and each column of $S$ represents my sources. $X$ is the data I observe.

If the columns of $S$ are independent and Gaussian, will the components of PCA be extremely similar to that of ICA? Is this the only requirement for the two methods to coincide?

Can someone provide an example of this being true when the $cov(X)$ isn't diagonal?

• I'd love to see a mathematical answer here -- something that starts from a derivation and demonstrates concretely where the two diverge and in what general case they are equivalent. – jvriesem Nov 30 '16 at 18:15