For SVD decomposition, if $X = U \Sigma V^T$. Then for
$$XX^T=U\Sigma V^T V\Sigma U^T=U \Sigma I \Sigma U^T = U \Sigma^2 U^T = \Sigma^2 \ ?$$ I believe $U$ and $V$ are both orthonormal matrices so this should work. But could someone help point out the error I have made?
As it's definitely not true that all matrices are diagonals.