$\newcommand{\SOn}{\operatorname{SO}_n}$
$\newcommand{\On}{\operatorname{O}_n}$
$\newcommand{\Sym}{\operatorname{Sym}_n}$
$\newcommand{\Skew}{\operatorname{Skew}_n}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\al}{\alpha}$
$\newcommand{\id}{\operatorname{Id}}$
This question is a reference request.
Claim:
Let $A $ be a $n \times n$ real matrix with non-positive determinant.
Then there is a unique closest matrix $Q \in \SOn$ to $A$ (w.r.t the Euclidean Frobenius norm) if and only if the smallest singular value of $A$ is strictly smaller than the rest of the singular values.
Question: Is this claim "known"? Where can I find a reference for it?
Note that I am looking for a reference, not a proof. (I have a proof…)
Also, I am specifically asking for the distance minimizer in $\SOn$. If we replace $\SOn$ with $\On$, then the minimizer(s) is the orthogonal polar factor from polar decomposition. (and is unique whenever the matrix $A$ is invertible).
Best Answer
A casual google search reveals that the result was mentioned in Myronenko and Song (2009), On the closed-form solution of the rotation matrix arising in computer vision problems, arXiv:0904.1613v1:
However, as orthogonal Procrustes problem and its variants have almost been studied to death, I am certain that the result had been mentioned by other earlier papers.