Assume that $A = U \* S$ for $U$ orthogonal and $S$ diagonal, ordered and positive.


If I only know $A$, is it possible to obtain $U$ and $S$?


My first guess would be taking the singular value decomposition, since $U \* D \* I = A$ by construction, but, due to the non-uniqueness of the SVD, I don't know how to enforce this specific solution.



How to decompose a matrix into its orthogonal and diagonal parts (assuming it has that form)?
To turn Darsh Rajan's comment into an answer:


if $A = US$ with $U$ orthogonal, $S$ diagonal, then $A^TA = S^TU^TUS = S^2$, so we know $S^2$ (and this gives us a condition on which matrices can be expressed in this way). If we further assume $S$ is positive, then we can find $S$ uniquely, and so find $U = AS^{-1}$.


