MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 9 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.