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This question is a reference request. Consider the following claim.
Let $A $ be a $n \times n$ real matrix.
Then an orthogonal matrix $Q$ is a closest matrix to $A$ in $ O(n)$ (w.r.t the Euclidean Frobenius norm) if and only if it is an orthogonal polar factor of $A$, i.e. if $A=QP$, where $P=\sqrt{A^TA}$.
Question: Where can I find a reference for this claim? (I am quite sure this is well-known)
I ask for a reference for the coincidence of the set of orthogonal minimizers with the set of all possible factors in the polar decomposition.
Note that if $A$ is invertible, then there is a unique polar factor, which is also the unique minimizer. I specifically look for references which deal with the general (possibly singular) case.
(BTW, the more interesting direction is why every minimizer is a polar factor).
I tried to look e.g. in this book, but found no mention of this.