# What's the best orthonormal matrix to align two matrices in the operator norm sense?

Let $A,B \in R^{n\times r}$ with $A^\top B$ invertible. It is known that $$UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F},$$ where $USV^\top$ is the SVD of $A^\top B$ and $\mathcal{O}^{r\times r}$ means the set of $r\times r$ orthonormal matrices.

However, if I change the metric from Frobenius norm to operator norm, what is the best orthonormal matrix?

In other words, what's $R$ that attains the minimum of the following? $$\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{op}.$$

It seems that the two rotation matrices are not the same (for Frobenius and for operator norm). If this is true, what can we say about $$\|AUV^\top-B\|_\mathrm{op}$$ and how worse is it compared with the optimal one?

## 1 Answer

The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational Mathematics, 2(4), pp 393–405, 1994, which actually looks at solving this "Procrustes problem" in Schatten-p norms.