Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection of $X$ onto the unitary matrices? In other words, which unitary matrix $U$ minimizes the Frobenius distance $\|U - X\|_F$?
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$\begingroup$ Related: mathoverflow.net/questions/86539/… $\endgroup$– Yoav KallusCommented Nov 18, 2019 at 3:39
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1$\begingroup$ Same answer applies: if X = WSV is the svd, U = WV is the projection. $\endgroup$– Yoav KallusCommented Nov 18, 2019 at 3:40
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$\begingroup$ Hi @YoavKallus, thanks for your response. I looked at the text you suggested - their theorem is in terms of polar decomposition, not SVDs. How did you pull out the SVD result? I had already reduced the problem to the case when $X$ is diagonal, using unitary invariance of the Frobenius norm. Intuitively, the identity matrix is the optimal choice in this case, which leads to your result. How do you prove that the identity matrix is the best one when $X$ is diagonal? $\endgroup$– GautamCommented Nov 18, 2019 at 6:16
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1$\begingroup$ You already have the answer in your own post: for the case of a real nonnegative diagonal matrix, think of the projection to the less strict constraint of normalized rows. If this already lends you in the stricter constraint of unitarity, then the result is also the projection to the stricter constraint. Also, you can get the polar decomposition from SVD and check that the answers agree. $\endgroup$– Yoav KallusCommented Nov 18, 2019 at 6:53
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1$\begingroup$ Have you seen Kahan's note? $\endgroup$– J. M. isn't a mathematicianCommented Nov 18, 2019 at 9:50
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