Derive necessary conditions to find an approximation to A: $min_{u\in R^{m}} ||A - uv^{T}||_{F}$

Question

Derive and establish necessary conditions for the following statement.

Given an m-by-n real matrix $A$ and unit vector $v \in \mathbb{R}^{n}$, find $u \in \mathbb{R}^{m}$, so that the rank-1 matrix $uv^{T}$ is a best approximation to $A$: $min_{u\in R^{m}} ||A - uv^{T}||_{F}$.

We can assume that m, n, rank(A) are greater than two, and that there exists a best approximation, $u_{0}v^{T}$.

I'm not sure where to take this question, so I'd appreciate any help.

Answer 1

Let $f(u)||A-uv^T||^2=tr(AA^T-2uv^TA^T+uu^T)$. Its derivative is

$Df_u:h\in\mathbb{R}^m\rightarrow tr(-2hv^TA^T+2hu^T)$.

If $f$ admits a minimum in $u$, then, for every $h$, $tr(-hv^TA^T+hu^T)=0$, that implies $-v^TA^T+u^T=0$, or $u=Av$.