Let $X=\lambda\_0u\_0v\_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda\_0\in\mathbb{R}$, $u\_0,v\_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\arg\min\_{\substack{Y:Y=aU\\U\in O(n)\\a\in\mathbb{R}}}\|Y-X\|\_F^2$$
where $U$ is orthogonal and $\|\|\_F$ is the Frobenius norm. In other words, what is the best scaled orthogonal approximation of rank 1 matrix?



Best orthogonal approximation of rank 1 matrix
As suggested by Federico Poloni in a comment, it suffices via the SVD to consider the case when $X$ has $x\_{1,1} > 0$ and $x\_{i,j} = 0$ when $(i,j) \neq (1,1)$. Then for any orthogonal matrix $U$ with columns $\mathbf{u}\_1,\ldots,\mathbf{u}\_n$ and scalar $a \in \mathbb{R}$ we have
\begin{align\*}
 \|X - aU\|\_F^2 & = \|x\_{1,1}\mathbf{e}\_1 - a\mathbf{u}\_1\|^2 + \sum\_{j=2}^n\|a\mathbf{u}\_j\|^2 \\
 & = \big(x\_{1,1}^2 - 2ax\_{1,1}u\_{1,1} + a^2\big) + \big(\sum\_{j=2}^na^2\big) \\
 & = x\_{1,1}^2 - 2ax\_{1,1}u\_{1,1} + na^2,
\end{align\*}
where $\mathbf{e}\_1 \in \mathbb{R}^n$ is the first standard basis vector. There is only one term in this expression that depends on $U$ at all, so (for any given value of $a$) it is minimized when we make $u\_{1,1}$ as large as possible (i.e., $u\_{1,1} = 1$). The remaining columns of $U$ are arbitrary (but of course must be orthogonal to the first column $\mathbf{e}\_1$).


Our goal is then to minimize
\begin{align\*}
 x\_{1,1}^2 - 2ax\_{1,1} + na^2,
\end{align\*}
which is a quadratic in $a$ with vertex (i.e., minimum) at $a = x\_{1,1}/n$.


