I have the square matrix equation (all matrices of dimension $n \times n$):
$G \Lambda G' = \Sigma$ where $\Lambda$ and $\Sigma$ are diagonal (with positive values on the diagonal). Under what sufficient conditions are the only solutions for $G$ equal
$G = \Sigma^{1/2} V \Lambda^{-1/2}$ for a $V$ such that $VV' = I$?