Actual moon runes: $\large \begin{align} A &= U\Sigma V^\intercal \\ A^\intercal A &= \left( U\Sigma V^\intercal \right)\left( U\Sigma V^\intercal \right) \\ &= U\Sigma^\intercal \left( U^\intercal U \right) \Sigma V^\intercal \\ &= V\Sigma^\intercal I \Sigma V^\intercal \\ &= V\pa{\Sigma^\intercal \Sigma} V^{-1} \\ \\ AA^\intercal &= U\Sigma V^\intercal \pa{U\Sigma V^\intercal}^\intercal \\ &=U\Sigma V^\intercal V \Sigma^\intercal U^\intercal \\ &= U\left( \Sigma \Sigma^\intercal \right)U^{-1} \\ &= \op{diagonalization of } AA^\intercal \\ \\ A^\intercal &= V \Sigma^\intercal U^\intercal \end{align} $