Let $A$ be a positive-definite matrix and $D$ be a diagonal matrix with positive entries. How can I determine a positive-definite matrix $X$ such that $X D X = A$?
If $D$ is the identity, then I can write the eigendecomposition $A = P M P^\intercal$ and let $X = P \sqrt{M} P^\intercal$, but how does it generalize?
Easily: write the equation as $(\Lambda^{-1/2}~A~\Lambda^{-1/2})^2 = \Lambda^{-1/2} \Sigma~\Lambda^{-1/2}$