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Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:

\begin{equation} \|R-M\|_F \end{equation}

Is there a closed form solution for $R$, or is it possible to express $R$ as the solution to a linear system? I would like to avoid gradient descent if possible.

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Let $M=U\Sigma V$ be the singular value decomposition of $M$, then $R=UV$. If you want $R$ to be a proper rotation (i.e. $\det R=1$) and $UV$ is not, replace the singular vector $\mathbf{u}_3$ associated with the smallest singular value of $M$ with $-\mathbf{u}_3$ in the $U$ matrix. An appropriate reference for this answer is:

N. J. Higham. Matrix nearness problems and applications. In M. J. C. Gover and S. Barnett, editors, Applications of Matrix Theory, pages 1–27. Oxford University Press, 1989.

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  • $\begingroup$ What if the distance measure is operator norm? Thank you! $\endgroup$
    – Wuchen
    Commented Sep 7, 2017 at 20:05
  • $\begingroup$ @Wuchen The same is true for the L-2 operator norm. Simply note that multiplying on the left by $U^{-1}$ and on the right by $V^{-1}$ leaves the problem (distances and constraints) invariant. Therefore, the problem is reduced to the case of diagonal matrices, which is straightforward. $\endgroup$
    – Yoav Kallus
    Commented Sep 16, 2017 at 20:22
  • $\begingroup$ @YoavKallus I don't find the case of diagonal matrices straightforward. It's true that replacing $\Sigma$ with $I$ is a feasible solution, but it is not that clear that it is the optimal one. $\endgroup$
    – Federico Poloni
    Commented Nov 18, 2019 at 12:43
  • $\begingroup$ @FedericoPoloni Since the identity is the nearest matrix with normalized rows to any nonnegative diagonal matrix and also happens to be orthogonal, it is also the nearest orthogonal matrix. $\endgroup$
    – Yoav Kallus
    Commented Nov 18, 2019 at 14:01
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    $\begingroup$ @FedericoPoloni Ah, sorry I misunderstood your question. The op norm distance to any other row-normalized matrix Q from S is at least the L-2 distance of the row of S containing the sv farthest from 1 to the corresponding row in Q. Therefore, it cannot be smaller than when Q is I (but can be equally small). See Kahan's note linked in the recent question on the unitary version (people.eecs.berkeley.edu/~wkahan/Math128/NearestQ.pdf). $\endgroup$
    – Yoav Kallus
    Commented Nov 18, 2019 at 14:52

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