If we have a diagonal covariance matrix does that guarantee independency?
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No, not in general. A covariance of zero between two random variables does not necessarily imply that they are independent. However, the statement is true if the variables are normally distributed.
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$\begingroup$ In fact, zero covariance does not imply independence even for normally distributed variables. See here. Rather, if two variables are from a multivariable normal distribution and are zero covariance, it implies independence. $\endgroup$ Feb 4, 2021 at 22:42