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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.

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    $\begingroup$ Isn't the diagonal of S just composed of the norms of the columns of A? $\endgroup$
    – Darsh Ranjan
    Commented Jul 18, 2018 at 15:25
  • $\begingroup$ You are correct, sorry for not noticing that before. $\endgroup$
    – HesterJ
    Commented Jul 18, 2018 at 15:31
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    $\begingroup$ Darsh's comment solves your problem, but it looks like you could need the polar decomposition in the near future. $\endgroup$
    – Federico Poloni
    Commented Jul 18, 2018 at 16:53

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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 $U = AS^{-1}$.

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