Is $\Sigma = A^T\Sigma' A$ the same as $\Sigma = A\Sigma' A^T$ where $\Sigma$ is a symmetrical matrix and $\Sigma'$ is a diagonal matrix? If so, how can I prove this?
My textbook explains that a multivariate normal distribution can be written in terms of a diagonalised form of covariance $\Sigma$, using $\Sigma = A^T\Sigma' A$, by the process of singular value decomposition. However, other online sources explaining covariance decomposition mainly use $\Sigma = A^T\Sigma' A$ where it is described as an eigendecomposition. I'm confused as to whether they are the same thing.