Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$


\begin{equation\*}
\arg\min\_{x\in\mathbb{C}^n} \|uv^\*+vu^\*-xx^\*\|^2\_F
\end{equation\*}


F here denotes Frobenius norm.



Best rank one approximation
We assume that the system $u,v$ is $\mathbb{C}$-free. Let $w=uv^\*+vu^\*$. Note that if $X$ is a solution then $\exp(i\theta)X$ is also a solution. We study the minimum of $trace((w-XX^\*)^2)=trace(w^2+XX^\*XX^\*-wXX^\*-XX^\*w)=trace(w^2)+trace((X^\*X)^2)-2trace(wXX^\*)$,


that is we study the minimum of $\phi(X)=||X||^4-2X^\*wX$. The hermitian matrix $w$ has rank $2$ and its eigenvalues are $0$ ($n-2$ times), a negative eigenvalue and a positive eigenvalue $\lambda$ (take an orthonormal basis $e\_1,e\_2$ of the plane $[u,v]$ s.t $e\_1$ and $u$ are collinear). Thus $\phi(X)\geq \psi(||X||^2)=||X||^4-2\lambda||X||^2$. Then $\inf(\psi(z))=\psi(\lambda)=-\lambda^2$ and $\phi(X)\geq -\lambda^2$. We can reach this bound: $X$ is an eigenvector of $w$ associated to $\lambda$ and $||X||^2=\lambda$. Note that the previous conditions define $X$ up to a factor $\exp(i\theta)$.


